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We propose a new method for constructing nullnorms on bounded lattices based on the simultaneous use of a closure operator and an interior operator. This method requires as input a given underlying t-conorm on a join sub-semilattice (depending on the closure operator), a given underlying t-norm on a meet sub-semilattice (depending on the interior operator) and an auxiliary mapping. It is shown that a nullnorm constructed by Karacal et al. in 2015 and a nullnorm constructed by Ertugrul in 2018 can be retrieved as special cases. We also explain how to obtain the dual results.

In an intersection join, we are given t sets R 1 , . . . , R t of axis-parallel rectangles in R d , where d ≥ 1 and t ≥ 2 are constants, and a join topology which is a connected undirected graph G on vertices 1 , . . . , t . The result consists of tuples ( r 1 , . . . , r t ) ∈ R 1 × . . . × R t where r i ∩ r j ≠ ∅ for all i , j connected in G. A structure is feasible if it stores O ˜ ( n ) words, supports an update in O ˜ ( 1 ) amortized time, and can enumerate the join result with an O ˜ ( 1 ) delay, where n = ∑ i | R i | and O ˜ ( . ) hides a polylog n factor. We provide a dichotomy as to when feasible structures exist: they do when t = 2 or d = 1 ; subject to the OMv-conjecture, they do not exist when t ≥ 3 and d ≥ 2 , regardless of the join topology.

Given a graph H and an integer k ≥ 1 , the Gallai–Ramsey number G R k ( H ) is defined to be the minimum integer n such that every k -edge coloring of the complete graph K n contains either a rainbow (all different colored) triangle or a monochromatic copy of H . In this paper, we study Gallai–Ramsey numbers for graphs with chromatic number three such as K m for m ≥ 2 , where K m is a kipas with m + 1 vertices obtained from the join of K 1 and P m , and a class of graphs with five vertices, denoted by ℋ . We first study the general lower bound of such graphs and propose a conjecture for the exact value of G R k ( K m ) . Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of G R k ( K 4 ) for k ≥ 1 . Our outcomes not only indicate that the conjecture on G R k ( K m ) is true for m ≤ 4 , but also imply several results on G R k ( H ) for some H ∈ ℋ which are proved individually in different papers.

Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in different fields such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we defined two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the first and second Zagreb indices and first reformulated Zagreb index for these new products.

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α =1/2. Several realization problems are also generated in this paper.

Let G be a connected graph. A subset S of V (G) is a connected co-independent hop dominating set in G if the subgraph induced by S is connected and V (G)\S is an independent set where for each v ∈ V (G)\S, there exists a vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of such an S is called the connected co-independent hop domination number of G. This paper presents the characterizations of the connected co-independent hop dominating sets in the join, corona and lexicographic product of two graphs. It also discusses the corresponding connected co-independent hop domination numbers of the aforementioned graphs.

A set S ⊆ V (G) is a restrained strong resolving dominating set in G if S is a strongresolving dominating set in G and S = V (G) or ⟨V (G) \ S⟩ has no isolated vertex. The restrained strong resolving domination number of G, denoted by γrsR(G), is the smallest cardinality of a restrained strong resolving dominating set in G. In this paper, we present characterizations of the restrained strong resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the exact value of the restrained strong resolving domination number of each of these graphs.

Let G be a connected simple graph with characteristic polynomial P G ( x ) . The irreducibility of P G ( x ) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P 2 n + 1 − 2 ( n ≥ 1 ) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.

Knowledge graph completion(KGC) has attracted increasing attention in recent years, aiming at complementing missing relationships between entities in a Knowledge Graph(KG). While the existing KGC approaches utilizing the knowledge within KG could only complement a very limited number of missing relations, more and more approaches tend to study the completion of the multi-relationship knowledge graph. However, the existing completion methods of multi-relation knowledge graph regard knowledge graph as an undirected graph, which ignores the directionality of knowledge graph, so that the potential characteristics of multi-relation cannot be learned. Besides, most algorithms fail to explore the local information of knowledge because they ignore the different importance of entity adjacencies. In this paper, we propose to use local information fusion to join the entity and its adjacency relation, to acquiring the multi-relation representation. In addition, we try to specify distinct weights to model the direction of the relationship and apply the attention mechanism between entity nodes to obtain local information between entity nodes. Experiments conducted on three benchmark datasets and a medical domain knowledge graph dataset that we collect demonstrate the effectiveness of the proposed framework.

Two of the most studied graph labelings are the types of harmonious and graceful. A harmonious labeling of a graph of size m and order n, is an injective assignment of nonnegative integers smaller than m, such that the weights of the edges, which are defined as the sum of the labels of the end-vertices, are distinct consecutive integers after reducing modulo m. When n = m + 1, exactly two vertices of the graph have the same label. An α-labeling of a tree of size m is a bijective assignment of nonnegative integers, not larger than m, such that the labels on one stable set are smaller than the labels on the other stable set, and the weights of the edges, which are defined as the absolute difference of the labels of the end-vertices, are all distinct; this is the most restrictive type of graceful labeling. Even when these labelings are significantly different in their definitions of the weight, for certain kinds of graphs, there is a deep connection between harmonious and α-labelings. We present new families of harmoniously labeled graphs built on α-labeled trees. Among these new results there are three families of trees, the kth power of the path Pn, the join of a graph G and tK1 where G is a graph that admits a more restrictive type of harmonious labeling and its order is different of its size by at most one unit. We also prove the existence of two families of disconnected harmonius graphs: Kn, m ∪ K1, m − 1 and G ∪ T, where G is a unicyclic graph and T is a tree built with α-trees. In addition, we show that almost all trees admit a harmonious labeling.

We study the cross-join pairs in the cycles of linear feedback shift registers whose characteristic polynomials are of the form $l(x) = p_{1}(x)p_{2}(x)\cdots p_{k}(x)$ , where $p_{i}(x), 1\leq i\leq k$ are primitive polynomials of coprime degrees. Firstly, we use Coppersmith et al.’s generating function theory to derive the lower and upper bounds for the number of cross-join pairs. Then we design an algorithm to generate these cross-join pairs. The algorithm requires a preparatory phase which costs $O(2^{n'})$ time where $n'$ is the largest degree of $p_{i}(x), 1\leq i\leq k$ , and after that it costs only $O(n^{3})$ time to generate one cross-join pair where $n$ is the degree of $l(x)$ . We also consider a special class of cross-join pairs, for which the preparatory phase costs only $O(2^{n''})$ time where $n''$ is the second-largest degree of $p_{i}(x), 1\leq i\leq k$ . The number of these special cross-join pairs is about $\frac {1}{12}2^{2n-n'}$ . We present some experimental results, which validate our analysis and demonstrate the efficiencies of the algorithms. These cross-join pairs can be used in the cross-joining method to construct de Bruijn sequences.

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth., Comment: 12 pages, author accepted manuscript, to appear in Combinatorics, Probability and Computing

We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Mart\'inez-P\'erez and L. Paris. More generally, let $\Gamma$ be a finite simplicial graph with all edges labelled by positive even integers and $A_\Gamma$ be its associated Artin group; our new proof implies that if $A_T$ is poly-free (resp. normally poly-free) for every clique $T$ in $\Gamma$, then $A_\Gamma$ is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell--Jones Conjecture for even Artin groups. In particular, we show that if $A_\Gamma$ is an even Artin group such that each clique in $\Gamma$ either has at most 3 vertices, has all of its labels at least $6$, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by $2$), then $A_\Gamma$ satisfies the Farrell--Jones Conjecture. In addition, our methods enables us to obtain results for general Artin groups., Comment: 13 pages; fixed a small gap in Section 2 found by the referee, typo corrections; to appear in J. Group Theory

In this work, we investigate upper limits on the global escape fraction of ionizing photons ($f_{\rm esc/global}^{\rm abs}$) from a sample of galaxies probed for Lyman-continuum (LyC) emission characterized as non-LyC and LyC leakers. We present a sample of nine clean non-contaminated (by low-redshift interlopers, CCD problems, and internal reflections of the instrument) galaxies that do not show significant (>3σ) LyC flux in the range 880

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between some vertices. In inverse fuzzy graph, the membership value of edges are greater than or equal to the minimum of the membership value of the corresponding vertices. In inverse fuzzy mixed graph (IFMG), directed and undirected relations exist between some vertices and it can be used to analyze many graphical problems of real life such that the membership values of edges are greater than or equal to the minimum of the membership value of the corresponding pair of vertices. In this article, the concept of IFMG is introduced first with some of its properties. Then some isomorphic properties are studied and complement of IFMG is given. Different types of operations like union, intersection, product and join between two IFMGs are defined and investigated some of their related results. An algorithm of the proposed method is executed to identify some vertices. An application is depicted using the concept IFMG to examine the order of vertices in a social network group according to communication gaps.

The rankable and the compressible sets have been studied for more than a quarter of a century. We ask whether these classes are closed under the most important boolean and other operations. We study this question for both polynomial-time and recursion-theoretic compression and ranking, and for almost every case arrive at a Closed, a Not-Closed, or a Closed-Iff-Well-Known-Complexity-Classes-Collapse result. Although compression and ranking classes are capturing something quite natural about the structure of sets, it turns out that these classes are quite fragile with respect to closure properties, and many fail to possess even the most basic of closure properties. For example, we show that with respect to the join (aka disjoint union) operation: the P-rankable sets are not closed, whether the semistrongly P-rankable sets are closed is closely linked to whether P = UP ∩ coUP , and the strongly P-rankable sets are closed.

Hodes (2021) “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart (see Plotkin & Sterling 1986). This paper continues that project, addressing some familiar classical strengthenings of K (D, T, K4, KB, K5, Dio (the Diodorian strengthening of K) and GL), and their intuitionistic counterparts (see Plotkin & Sterling 1986 for some of these counterparts). Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Hodes (2021). For the systems associated with the intuitionistic counterparts of D and T, these rules are “pure one-step”: their schematic formulations does not use □ or ♢. For the systems associated with the intuitionistic counterparts of K4, etc., these rules meet these conditions: neither □ nor ♢ is iterated; none use both □ and ♢. The join of the two systems associated with each of these familiar logics is the full one-step system for that intuitionistic logic. And further “blended” intuitionistic systems arise from joining these systems in various ways. Adding the 0-version of Excluded Middle to their intuitionistic counterparts yields the one-step systems corresponding to the familiar classical logics. Each proof-theoretic system defines a consequence relation in the obvious way. Section 10 examines inclusions between these consequence relations. Section 11 associates each of the above consequence relations with an appropriate class of models, and proves them sound with respect to their appropriate class. This allows proofs of some failures of inclusion between consequence relations. (Sections 10 and 11 provide an exhaustive study of a variety of intuitionistic modal logics.) Section 12 proves that the each consequence relation is complete or (for those corresponding to GL) weakly complete, that relative to its appropriate class of models. The Appendix presents three further results about some of the intuitionistic consequence relations discussed in the body of the paper.

A set S ⊆ V(G) of a (simple) undirected graph G is a locating-dominating set of G if for each v ∈ V(G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S= NG(y)∩S for any distinct vertices x and y in V(G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γsl(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs.

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ V (G) is a perfect hop dominating set of G if for every v ∈ V (G) \ S, there is exactly one vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of a perfect hop dominating set of G is called the perfect hop domination number of G, denoted by γph(G). A perfect hop dominating set S ⊆ V (G) is called a total perfect hop dominating set of G if for every v ∈ V (G), there is exactly one vertex u ∈ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by γtph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality γtph(G) is referred to as a γtph-set of G. In this paper, we characterize the total perfect hop dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding total perfect hop domination number.

In this paper, the total dr-power domination number of graphs such as complete bipartite graph, generalized fan and generalized wheel are obtained. The forcing total dr-power domination number of graphs resulting from some binary operations such as join, corona and lexicographic product of graphs were determined.

Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set in G if, for any distinct vertices u, w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. If G has a 2-resolving set, we denote the least size of a 2-resolving set by dim2(G), the 2-metric dimension of G. A 2-resolving set of size dim2(G) is called a 2-metric basis for G. This study deals with the concept of 2-resolving set of a graph. It characterizes the 2-resolving set in the join and corona of graphs and determine theexact values of the 2-metric dimension of these graphs.

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

A set S of vertices in a connected graph G is a resolving hop dominating set of G if S is a resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the resolving hop domination number of G. This paper presents the characterizations of the resolving hop dominating sets in the join, corona and lexicographic product of two graphs and determines the exact values of their corresponding resolving hop domination number.

Fuzzy signatures have been used for various applications, including medical diagnosis, communication of robots based on intention guessing, and residential building evaluation. However, there exist many questions about this research topic which have not been addressed in the literature. One of the most important aims is to formally define a family of fuzzy signatures from which an algebraic structure can be obtained allowing to make computations among fuzzy signatures. This paper studies this family and defines suitable meet and join operators satisfying the properties of a lattice as an algebraic structure. A partial ordering relation, the least and greatest elements are also defined on the family of fuzzy signatures. As a consequence, fuzzy signatures can be used as truth values of fuzzy sets, which provides a great level of representativity, completely different from the interval-valued and type-2 fuzzy sets, among others.

Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real $$\alpha \in [0,1]$$ , the matrix $$A_{\alpha }(G)$$ is defined as $$A_{\alpha }(G) = \alpha D(G) + (1- \alpha ) A(G).$$ The eigenvalues of the matrix $$A_{\alpha }(G)$$ form the $$A_{\alpha }$$ -spectrum of G. Let $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs $$G_1$$ and $$G_2$$ , respectively. In this paper, we compute the $$A_{\alpha }$$ -spectra of $$G_1 {\dot{\vee }} G_2$$ , $$G_1 {\underline{\vee }} G_2$$ , $$G_1 \langle \text {v} \rangle G_2$$ and $$G_1 \langle \text {e} \rangle G_2$$ for a regular graph $$G_1$$ and an arbitrary graph $$G_2$$ in terms of their $$A_{\alpha }$$ -eigenvalues. As an application of these results, we construct infinitely many pairs of $$A_{\alpha }$$ -cospectral graphs.

Today, real-world problems modeling is the first step in controlling, analyzing, and optimizing them. One of the applied techniques for modeling some of these problems is the queueing theory. Usually, the conditions such as lack of space, feedback, admission limits, etc. are the inseparable parts of these problems. This paper deals with modeling and analyzing the steady-state behavior of an $$M^{X} /G/ 1$$ retrial queueing system with two phases of heterogeneous services and general retrial time. The arriving batches join the system with dependent admission due to the server state. If the customers find the server busy, they join the orbit to repeat their request. Although, the first phase of service is essential for all customers, any customer has three options after the completion of the $$i$$ -th phase $$\left( {i = 1,2} \right)$$ . They may take the $$\left( {i + 1} \right)$$ -th phase of service with probability $${\uptheta }_{{\text{i}}}$$ , otherwise, return the orbit with probability $$p_{i} \left( {1 - \theta_{i} } \right)$$ or leave the system with probability $${ }\left( {1 - p_{i} } \right)\left( {1 - \theta_{i} } \right)$$ . In this paper, after finding the steady-state distributions and the probability generating functions of the system and orbit size, some important performance measures are found. Then, the sensitivity analysis of performance measures and cost rate is done concerning arrival/retrial rates, feedback probabilities, and state-dependent admission in a telecommunication system.

Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.

Let G be an edge-colored connected graph. A path P in G is called $$\ell $$ -rainbow if no two edges of the same color have fewer than $$\ell $$ edges between them on P. The graph G is called $$(k,\ell )$$ -rainbow connected if every pair of distinct vertices of G are connected by k pairwise internally vertex-disjoint $$\ell $$ -rainbow paths in G. For a k-connected graph G, the minimum number of colors needed to make G $$(k,\ell )$$ -rainbow connected is called the $$(k,\ell )$$ -rainbow connection number of G and denoted by $$rc_{k,\ell }(G)$$ . In this paper, we prove that $$rc_{1,2}(G)\le 5$$ for any 2-connected graph G and provide a counter example which overturns a conjecture posed by Chartrand et al. in 2016. Considering graph operations, we study the (1, 2)-rainbow connection numbers for the join, the Cartesian, strong and lexicographic products of graphs, giving the sharp upper bounds for nearly all graph products. In addition, we find some basic properties of the $$(k,\ell )$$ -rainbow connection number and determine the values of $$rc_{1,\ell }(G)$$ where G is a cube or a permutation graph of a nontrivial traceable graph.

Given a finite group G, the permutability graph of non-normal subgroups $$\Gamma _{N}(G)$$ is given by all proper non-normal subgroups of G as vertex set and two vertices H and K are joined if $$HK = KH$$ . Rajkumar and Devi generalized $$\Gamma _{N}(G)$$ to the permutability graph of subgroups $$\Gamma (G)$$ , extending the vertex set to all proper subgroups of G (removing the assumption of being non-normal) and keeping the same criterion to join two vertices. We consider a natural counterpart for $$\Gamma (G)$$ and $$\Gamma _{N}(G)$$ , that is, the subgroups lattice $$\mathrm {L}(G)$$ of G, and introduce the non-permutability graph of subgroups $$\Gamma _{\mathrm {L}(G)}$$ ; its vertices are now given by the set $$\mathrm {L}(G)-{\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))$$ , where $${\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))$$ denotes the smallest sublattice of $$\mathrm {L}(G)$$ containing all permutable subgroups of G, and we join two vertices H, K of $$\Gamma _{\mathrm {L}(G)}$$ if and only if $$HK\ne KH$$ . We study classical invariants of $$\Gamma _{\mathrm {L}(G)}$$ , showing generalizations of previous results in the literature.

A natural way to interpret a cellular automaton (CA) is as a mechanism that computes, in a distributed way, some function f. In other words, from a computer science point of view, CAs can be seen as distributed systems where the cells of the CAs are nodes of a network linked by communication channels. A classic measure of efficiency in such distributed systems is the number of bits exchanged during the computation process. A typical approach is to look for bottlenecks: channels through which the nature of the function f forces the exchange of a significant number of bits. In practice, a widely used way to understand such congestion phenomena is to partition the system into two subsystems and try to find bounds for the number of bits that must pass through the channels that join them. Finding these bounds is the focus of communication complexity theory. Measuring the communication complexity of some problems induced by a CA $$\phi$$ turned out to be tremendously useful to give clues regarding the intrinsic universality of $$\phi$$ (a CA is said to be intrinsically universal if it is capable of emulating any other). In fact, there exist particular problems $$\mathrm {P}$$ ’s for which the following key property holds: if $$\phi$$ is intrinsically universal, then the communication complexity of $$\mathrm {P}(\phi )$$ must be maximal. In this tutorial, we intend to explain the connections that were found, through a series of papers, between intrinsic universality and communication complexity in CAs.

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.

For a commutative ring R with unity ( $$1\ne 0$$ ), the zero-divisor graph of R is a simple graph with vertices as elements of $$Z(R)^{*}=Z(R)\setminus \{ 0 \}$$ , where Z(R) is the set of zero-divisors of R and two distinct vertices are adjacent whenever their product is zero. An algorithm is presented to create a zero-divisor graph for the ring of Gaussian integers modulo $$2^{n}$$ for $$n\ge 1$$ . The zero-divisor graph $$\Gamma (\mathbb {Z}_{2^{n}}[i])$$ can be expressed as a generalized join graph $$G[G_{1}, \dots , G_{j}]$$ , where $$G_{i}$$ is either a complete graph (including loops) or its complement and G is the compressed zero-divisor graph of $$\mathbb {Z}_{2^{2n}}$$ . Next, we show that the number of isomorphisms between the zero-divisor graphs for the ring of Gaussian integers modulo $$2^{n}$$ and the ring of integers modulo $$2^{2n}$$ is equal to $$\prod _{j=1}^{2(n-1)}2^{j}!$$ .

Quite recently, Wu and Liu (2020) [11] raised an open problem when they discussed the relationships between generalized rough sets and topologies. Said precisely, each binary relation generates a topology through the lower rough approximation operator, then for two binary relations on the same set, is there a sufficient and necessary condition such that the union of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. In this note, we give a positive answer to this problem and provide a union-join condition. Furthermore, note that the main reason that the expected identity hold not consists in the fact that the union of two topologies is not a topology, in general. Then replacing the union of two topologies with the supremum of them, we give a sufficient and necessary condition (supremum-join condition) such that the supremum of generated topologies by two binary relations, is identical with the generated topology by the join of two binary relations. We verify that the supremum-join condition is much simpler than the union-join condition. At last, we give a characterization on discrete topology generated by a binary relation using the sober separation.

Phylogenetic analysis is used to analyze the evolution of species according to the characteristics of biological sequences. The analytical results are generally represented by phylogenetic trees. NJ (neighbor joining) is a frequently used algorithm for constructing phylogenetic trees because of its few assumptions, fast operation, and high accuracy, and is based on the distance between taxa. It is known that NJ usually constructs different phylogenetic trees for the same dataset with differences in input order, which are known as “tied trees.” This article proposes an improved method of NJ, called ENJ (extended neighbor joining). The ENJ can join several (currently limited to three) nodes with the same minimum distance into a new node, rather than joining two nodes in one iteration, so it can construct triple phylogenetic trees. We have inferred the formulas for updating the distance values and calculating the branch lengths for the ENJ algorithm. We have tested the ENJ with simulated and real data. The experimental results show that, compared with other methods, the trees constructed by the ENJ have greater similarity to the initial trees, and the ENJ is much faster than the NJ algorithm. Moreover, we have constructed a phylogenetic tree for the novel coronavirus (COVID-19) and related coronaviruses by ENJ, which shows that COVID-19 and SARS-CoV are closer than other coronaviruses. Because it differs from the existing phylogenetic trees for those coronaviruses, we constructed a phylogenetic network for them. The network shows those species have had a reticulate evolution., Graphical Abstract, Accurately inferring evolutionary relationships between species can help humans to understand the evolutionary history and the mechanism. ENJ is an efficient and effective algorithm for constructing phylogenetic trees, which can construct triple phylogenetic trees, if necessary. The trees constructed by ENJ can better describe the original information and reflect the real evolutionary relationship of species.

There are several tools, like algebraic polynomials, topological indices, graph energies, etc., to investigate the structural dependence of different properties and activities of chemical structure...

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). The membership grades of T2FSs are T1FSs in interval [0, 1], which are referred to as fuzzy truth values (that is, functions from [0, 1] to [0, 1]). Recently, (convolution) operations on T2FSs have become a hot research topic, such as type-2 t-norms, type-2 aggregation operations, type-2 implications, and so on. So, the distributive laws between these convolution operations have become an interesting and natural research area. In this article, we investigate the distributive laws of convolution operations over meet-convolution and join-convolution (two basic operations on fuzzy truth values), respectively. At first, we consider whether the distributive laws hold in various subsets of the set of all fuzzy truth values, such as set-valued T2FSs, singletons, interval-valued T2FSs, and the set of all normal convex fuzzy truth values. Then, we discuss whether the distributive laws hold for some specific convolution operations. Finally, we give a necessary and sufficient condition under which join-convolution is distributive over meet-convolution, and similarly give a necessary and sufficient condition under which meet-convolution is distributive over join-convolution.

A set S ⊆ V ( G ) is a hop dominating set of G if for each v ∈ V ( G ) \ S , there exists w ∈ S such that d G ( v, w ) = 2. It is a global hop dominating set of G if it is a hop dominatingset of both G and the complement of G . The minimum cardinality of a global hop dominatingset of G , denoted by γ gh ( G ), is called the global hop domination number of G . In this paper, we study the concept of global hop domination in graphs resulting from some binary operations.

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

We prove that the symmetrized deleted join SymmDelJoin( $$\mathcal{K}$$ ) of a “balanced family” $$\mathcal{K}$$ = 〈Ki〉 =1 of collectively r-unavoidable subcomplexes of 2[m] is (m−r−1)-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result which is more conceptual and more general than previously known results. Already the case r = 2 of this result seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman’s discrete Morse theory.

Let Q(G), $${{\mathcal {D}}(G)}$$ and $${{\mathcal {D}}}^Q(G)={{\mathcal {D}}iag(Tr)} + {{\mathcal {D}}(G)}$$ be, respectively, the signless Laplacian matrix, the distance matrix and the distance signless Laplacian matrix of graph G, where $${{\mathcal {D}}iag(Tr)}$$ denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of Q(G) and $${{\mathcal {D}}}^Q(G)$$ will be denoted by $$q_{1} \ge q_{2} \ge \cdots \ge q_{n-1} \ge q_n$$ and $$\partial ^Q_1 \ge \partial ^Q_2 \ge \cdots \ge \partial ^Q_{n-1} \ge \partial ^Q_n$$ , respectively. A graph G which does not share its distance signless Laplacian spectrum with any other non-isomorphic graphs is said to be determined by its distance signless Laplacian spectrum. Characterizing graphs with respect to spectra of graph matrices is challenging. In literature, there are many graphs that are proved to be determined by the spectra of some graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix etc.). But there are much fewer graphs that are proved to be determined by the distance signless Laplacian spectrum. Namely, the path graph, the cycle graph, the complement of the path and the complement of the cycle are proved to be determined by the distance signless Laplacian spectra. In this paper, we establish Nordhaus–Gaddum-type results for the least signless Laplacian eigenvalue of graph G. Moreover, we prove that the join graph $$G\vee K_{q}$$ is determined by the distance singless Laplacian spectrum when G is a $$p-2$$ regular graph of order p. Finally, we show that the short kite graph and the complete split graph are determined by the distance signless Laplacian spectra. Our approach for characterizing these graphs with respect to distance signless Laplacian spectra is different from those given in literature.

We introduce a notion of color-criticality in the context of chromatic-choosability. We define a graph $G$ to be strong $k$-chromatic-choosable if $\chi(G) = k$ and every $(k-1)$-assignment for which $G$ is not list-colorable has the property that the lists are the same for all vertices. That is the usual coloring is, in some sense, the obstacle to list-coloring. We prove basic properties of strongly chromatic-choosable graphs such as chromatic-choosability and vertex-criticality, and we construct infinite families of strongly chromatic-choosable graphs. We derive a sufficient condition for the existence of at least two list colorings of strongly chromatic-choosable graphs and use it to show that: if $M$ is a strong $k$-chromatic-choosable graph with $|E(M)| \leq |V(M)|(k-2)$ and $H$ is a graph that contains a Hamilton path, $w_1, w_2, \ldots, w_m$, such that $w_i$ has at most $\rho \geq 1$ neighbors among $w_1, \ldots, w_{i-1}$, then $\chi_{\ell}(M \square H) \le k+ \rho - 1$. We show that this bound is sharp for all $\rho \ge 1$ by generalizing the theorem to apply to $H$ that are $(M,\rho)$-Cartesian accommodating which is a notion we define with the help of the list color function, $ P_{\ell}(G,k)$, the list analogue of the chromatic polynomial. We also use the list color function to determine the list chromatic number of certain star-like graphs: $\chi_{\ell}(M \square K_{1,s}) =$ $k \; \text{if } s < P_{\ell}(M,k)$, or $k+1 \; \text{if } s \geq P_{\ell}(M,k)$, where $M$ is a strong $k$-chromatic-choosable graph. We show that $ P_{\ell}(M,k)$ equals $P(M,k)$, the chromatic polynomial, when $M$ is an odd cycle, complete graph, or the join of an odd cycle with a complete graph., Comment: 27 pages

Center coloring Cc(G) is a kind of coloring that is to color the vertices of a graph G is such a way that if vertices have different distances from the center then they must receive different colors. Two adjacent vertices can receive the same color. In this paper we observed the behavior of the center coloring under several graph operations such as adding, removing, union, join, sequential join, Cartesian product, corona product of graphs and we get some bounds for center coloring number in these graph operations.

The huge amount of data enforces great pressure on the processing efficiency of database systems. By leveraging the in-situ computing ability of emerging nonvolatile memory, processing-in-memory (PIM) technology shows great potential in accelerating database operations against traditional architectures without data movement overheads. In this article, we introduce ReSQM, a novel ReCAM-based accelerator, which can dramatically reduce the response time of database systems. The key novelty of ReSQM is that some commonly used database queries that would be otherwise processed inefficiently in previous studies can be in-situ accomplished with massively high parallelism by exploiting the PIM-enabled ReCAM array. ReSQM supports some typical database queries (such as SELECTION, SORT, and JOIN) effectively based on the limited computational mode of the ReCAM array. ReSQM is also equipped with a series of hardware-algorithm co-designs to maximize efficiency. We present a new data mapping mechanism that allows enjoying in-situ in-memory computations for SELECTION operating upon intermediate results. We also develop a count-based ReCAM-specific algorithm to enable the in-memory sorting without any row swapping. The relational comparisons are integrated for accelerating inequality join by making a few modifications to the ReCAM cells with negligible hardware overhead. The experimental results show that ReSQM can improve the (energy) efficiency by $611\times $ ( $193\times $ ), $19\times $ ( $17\times $ ), $59\times $ ( $43\times $ ), and $307\times $ ( $181\times $ ) in comparison to a 10-core Intel Xeon E5-2630v4 processor for SELECTION, SORT, equi-join, and inequality join, respectively. In contrast to state-of-the-art CMOS-based CAM, GPU, FPGA, NDP, and PIM solutions, ReSQM can also offer $2.2\times 39\times $ speedups.

This brief proposes a join algorithm based on a top $k$ -sorter. The algorithm applies a filter to remove tuples having no potential matches with the aim of reducing the number of comparisons during join operations between two tables ( $T_{a},T_{b}$ ). Based on this algorithm, we designed a modular pipeline join architecture that comprises $k$ cascaded processing units. The resource consumption of the architecture is unaffected by the table size such that a join operation between tables with arbitrary sizes is supported. The join process includes a loop in which the $k$ “largest” tuples of the table with smaller size ( $T_{a}$ ) are identified to match and filter the tuples of the other table ( $T_{b}$ ). Each tuple of $T_{b}$ is compared with the $k$ largest tuples in turn and discarded if a “smaller” tuple is encountered. After each iterative cycle, the $k$ largest tuples of $T_{a}$ are also removed. The loop terminates once either of the two tables becomes empty. The experimental results show that the proposed join architecture is not only resource efficient but also achieves high throughput on a state-of-the-art Field Programmable Gate Array (FPGA).

In this paper, we introduce and investigate the concepts of semitotal k-fair domination and independent k-fair domination, where k is a positive integer. We also characterize the semitotal 1-fair dominating sets and independent k-fair dominating sets in the join, corona, lexicographic product, and Cartesian product of graphs and determine the exact value or sharp bounds of the corresponding semitotal 1-fair domination number and independent k-fair domination number.

Given two datasets (or tables) A and B and a search distance $$\epsilon$$ ϵ , the distance similarity join, denoted as $$A \ltimes _\epsilon B$$ A ⋉ ϵ B , finds the pairs of points ($$p_a$$ p a , $$p_b$$ p b ), where $$p_a \in A$$ p a ∈ A and $$p_b \in B$$ p b ∈ B , and such that the distance between $$p_a$$ p a and $$p_b$$ p b is $$\le \epsilon$$ ≤ ϵ . If $$A = B$$ A = B , then the similarity join is equivalent to a similarity self-join, denoted as $$A \bowtie _\epsilon A$$ A ⋈ ϵ A . We propose in this paper Heterogeneous Epsilon Grid Joins (HEGJoin), a heterogeneous CPU-GPU distance similarity join algorithm. Efficiently partitioning the work between the CPU and the GPU is a challenge. Indeed, the work partitioning strategy needs to consider the different characteristics and computational throughput of the processors (CPU and GPU), as well as the data-dependent nature of the similarity join that accounts in the overall execution time (e.g., the number of queries, their distribution, the dimensionality, etc.). In addition to HEGJoin, we design in this paper a dynamic and two static work partitioning strategies. We also propose a performance model for each static partitioning strategy to perform the distribution of the work between the processors. We evaluate the performance of all three partitioning methods by considering the execution time and the load imbalance between the CPU and GPU as performance metrics. HEGJoin achieves a speedup of up to $$5.46\times$$ 5.46 × ($$3.97\times$$ 3.97 × ) over the GPU-only (CPU-only) algorithms on our first test platform and up to $$1.97\times$$ 1.97 × ($$12.07\times$$ 12.07 × ) on our second test platform over the GPU-only (CPU-only) algorithms.

In this paper we introduce the notion of group contributions in TU-games and propose a new class of values which we call the class of k-lateral Shapley values. Most of the values for TU-games implicitly assume that players are independent in deciding to leave or join a coalition. However, in many real life situations players are bound by the decisions taken by their peers. This leads to the idea of group contributions where we consider the marginality of groups upto a certain size. We show that group contributions can play an important role in determining players’ shares in the total resource they generate. The proposed value has the flavor of egalitarianism within group contributions. We provide two characterizations of our values.

We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied.

Topological indices play a very important role in the mathematical chemistry. The topological indices are numerical parameters of a graph. The degree sequence is obtained by considering the set of vertex degree of a graph. Graph operators are the ones which are used to obtain another broader graphs. This paper attempts to find degree sequence of vertex–F join operation of graphs for some standard graphs.