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The paper examines various properties of D T - k -subgroup structures and addresses an open problem on the existence of topological group structures on the n -dimensional Khalimsky (K -, for brevity) topological space and Marcus–Wyse (M -, for short) topological plane. In particular, we obtain many types of totally k -disconnected or k -connected subgroups of a k -connected D T - k -group. Besides, we prove that each of the n -dimensional K -topological space and the M -topological plane cannot be a typical topological group. Unlike an existence of a D T - k -group structure of (S C k n , l , ∗) (see Proposition 4.7), we prove that neither of (S C K n , l , ∗) and (S C M l , ∗) is a topological group, where S C K n , l (respectively, S C M l ) is a simple closed K - (respectively, M -) topological curve with l elements in ℤ n (respectively, ℤ 2 ) and the operation "∗" is a special kind of binary operation for establishing a group structure of each of S C K n , l and S C M l . Finally, given a D T - k -group structure of (S C k 1 n 1 , l 1 × S C k 2 n 2 , l 2 , ∗) , we find several types of D T - k -subgroup structures of it (see Theorem 5.7). [ABSTRACT FROM AUTHOR]

The enhanced power graph of a group G is a simple undirected graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we obtain the Laplacian spectrum of the enhanced power graph of certain non-abelian groups, viz. semidihedral, dihedral and generalized quaternion. Also, we obtained the metric dimension and the resolving polynomial of the enhanced power graphs of these groups. At the final part of this paper, we study the distant properties and the detour distant properties, namely: closure, interior, distance degree sequence, eccentric subgraph of the enhanced power graph of semidihedral group, dihedral group and generalized quaternion group, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a newly derived signed graph called a coset component graph, based on cosets of subgroups of a group is introduced. Let G be a group and H be its subgroup. Then, the coset component graph of H in G , denoted by Γ cc , is a simple graph with the vertex set consisting of elements of G and two vertices say, a , b ∈ Γ cc are adjacent if either a H = b H or H a = H b. A coset component signed graph of H in G is a signed graph whose edges get the sign in accordance with their inclusion in the edge set of the corresponding coset component graph. The structure and important properties of the coset component signed graphs are determined in this paper. [ABSTRACT FROM AUTHOR]

Traditional two-color lights out games and their solutions have been studied extensively. This paper focuses on the existence, or lack there of, of solutions in more general lights out games, which take on a prime number of color states. The result in this paper generalizes the existence of solutions in the family of lights out games on a (p + 1) × n grid with p -colors where n ∈ N and p. [ABSTRACT FROM AUTHOR]

The concept of regular relations at sets and the concepts of some other relations derived from these, such as normal and conjugative relations, have been the subject of study by several researchers. The concepts of bi-conjugative relations, finite extension of bi-conjugative relations and weakly finite extension of bi-conjugative relations were introduced and analyzed by Romano. In this paper, as a continuation of previous research, the author introduces and analyzes the concept of strongly bi-conjugative relations and one of its finitely extension as a specific case of bi-conjugative relations. [ABSTRACT FROM AUTHOR]

In recent years, deep learning has revolutionized natural language processing (NLP) by enabling the development of models that can learn complex representations of language data, leading to significant improvements in performance across a wide range of NLP tasks. Deep learning models for NLP typically use large amounts of data to train deep neural networks, allowing them to learn the patterns and relationships in language data. This is in contrast to traditional NLP approaches, which rely on hand-engineered features and rules to perform NLP tasks. The ability of deep neural networks to learn hierarchical representations of language data, handle variable-length input sequences, and perform well on large datasets makes them well-suited for NLP applications. Driven by the exponential growth of textual data and the increasing demand for condensed, coherent, and informative summaries, text summarization has been a critical research area in the field of NLP. Applying deep learning to text summarization refers to the use of deep neural networks to perform text summarization tasks. In this survey, we begin with a review of fashionable text summarization tasks in recent years, including extractive, abstractive, multi-document, and so on. Next, we discuss most deep learning-based models and their experimental results on these tasks. The paper also covers datasets and data representation for summarization tasks. Finally, we delve into the opportunities and challenges associated with summarization tasks and their corresponding methodologies, aiming to inspire future research efforts to advance the field further. A goal of our survey is to explain how these methods differ in their requirements as understanding them is essential for choosing a technique suited for a specific setting. This survey aims to provide a comprehensive review of existing techniques, evaluation methodologies, and practical applications of automatic text summarization. [ABSTRACT FROM AUTHOR]

In this paper, the b -coloring problem is solved for every subdivision-edge neighborhood corona of paths, cycles, stars and complete graphs. In addition, some exact values and sharp bounds are established for the b -chromatic number of subdivision-edge neighborhood coronas of these graphs with any other graph. [ABSTRACT FROM AUTHOR]

An injective k -edge coloring of a graph G is a k -edge coloring φ of G such that φ (e 1) ≠ φ (e 3) for any three consecutive edges e 1 , e 2 and e 3 of a path or a 3-cycle. The injective edge chromatic index of G , denoted by χ i ′ (G) , is the minimum k such that G has an injective k -edge coloring. Li and Chen determined the injective edge chromatic index of the generalized Petersen graph P (n , k) for 1 ≤ k ≤ 2. In this paper, we consider the injective edge coloring of the generalized Petersen graph P (c k , k) and show that χ i ′ (P (c k , k)) ≤ 5 for k ≥ 3. [ABSTRACT FROM AUTHOR]

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let f be a map from V (G) to { 0 , 1 , ... , k − 1 } where k is an integer, 1 ≤ k ≤ | V (G) |. For each edge u v assign the label f (u) f (v) (mod  k). f is called a k-product cordial labeling if | v f (i) − v f (j) | ≤ 1 , and | e f (i) − e f (j) | ≤ 1 , i , j ∈ { 0 , 1 , ... , k − 1 } , where v f (x) and e f (x) denote the number of vertices and edges, respectively, labeled with x (x = 0 , 1 , ... , k − 1). A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling. [ABSTRACT FROM AUTHOR]

The eccentricity matrix ℰ (G) = ( u v) of a graph G is constructed from the distance matrix by keeping each row and each column only the largest distances with u v = d (u , v) , if d (u , v) = min { (u) , (v) } , 0 , otherwise , where d (u , v) is the distance between two vertices u and v , and (u) = max { d (u , v) | v ∈ V (G) } is the eccentricity of the vertex u. The ℰ -eigenvalues of G are those of its eccentricity matrix. In this paper, employing the well-known Cauchy Interlacing Theorem we give the following lower bounds for the second, the third and the fourth largest ℰ -eigenvalues by means of the diameter d of G : ξ 2 (G) ≥ − 1 , if d ≤ 2 ; α d , if d ≥ 3 , ξ 3 (G) ≥ − d , and ξ 4 (G) ≥ − 1 − 5 2 d , where α = 0. 3 1 1 1 + is the second largest root of x 3 − x 2 − 3 x + 1 = 0. Moreover, we further discuss the graphs achieving the above lower bounds. [ABSTRACT FROM AUTHOR]

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

A majority double Roman dominating function (MDRDF) on a graph G = (V , E) is a function f : V → { − 1 , + 1 , 2 , 3 } such that (i) every vertex v with f (v) = − 1 is adjacent to at least two vertices assigned with 2 or to at least one vertex w with f (w) = 3 , (ii) every vertex v with f (v) = 1 is adjacent to at least one vertex w with f (w) ≥ 2 and (iii) ∑ u ∈ N [ v ] f (u) ≥ 1 , for at least half of the vertices in G. The weight of an MDRDF is the sum of its function values over all vertices. The majority double Roman domination number of a graph G , denoted by γ MDR (G) , is defined as γ MDR (G) = min { w (f)   |   f   is an  MDRDF  of   G }. In this paper, we introduce and study the majority double Roman domination number on some classes of graphs. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and Nil (R) be the ideal of all nilpotent elements of R. Let (R) = { I : I be a nontrivial ideal of R and there exists a nontrivial ideal J such that I J ⊆ Nil (R) }. The nil-graph of ideals of R is defined as the graph N (R) whose vertex set is the set (R) and two distinct vertices I and J are adjacent if and only if I J ⊆ Nil (R). A subset of vertices S ⊆ V (G) resolves a graph G , and S is a resolving set of G , if every vertex is uniquely determined by its vector of distances to the vertices of S. In particular, for an ordered subset S = { v 1 , v 2 , ... , v k } of vertices in a connected graph G and a vertex v ∈ V (G) \ S of G , the metric representation of v with respect to S is the k -vector D (v | S) = (d (v , v 1) , d (v , v 2) , ... , d (v , v k)). The set S is a resolving set for G if D (u | S) = D (v | S) implies that u = v , for all pair of vertices, v , u ∈ V (G) \ S. A resolving set S of minimum cardinality is the metric basis for G , and the number of elements in the resolving set of minimum cardinality is the metric dimension of G. If D (u | S) ≠ D (v | S) for every pair u , v of adjacent vertices of G , then S is called a local metric set of G. The minimum k for which G has a local metric k -set is the local metric dimension of G , which is denoted by lmd (G). In this paper, we determine metric dimension and local metric dimension of nil-graph of ideals of commutative rings. [ABSTRACT FROM AUTHOR]

Given a graph G with no isolated vertices, let γ e v (G) , i e v (G) , β e v (G) , γ (G) , γ pr (G) and Γ pr (G) denote the ev-domination number, the independent ev-domination number, the upper independent ev-domination number, the domination number, the paired-domination number and the upper paired-domination number, respectively. It is known that γ e v (G) = i e v (G) = 1 2 γ pr (G) ≤ β e v (G). In this paper, we extend this inequality chain to involve the upper paired-domination number for arbitrary graphs G with no isolated vertices as well as the domination number for trees. Moreover, we show that recognizing well ev-covered graphs (i.e., graphs G with i e v (G) = β e v (G)) is co-NP-complete, solving an open problem posed by Boutrig and Chellali. [ABSTRACT FROM AUTHOR]

The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph. Entropy functions have been used successfully to capture different aspects of graph complexity. The generalized graph entropies result from applying information measures to a graph using various schemes for defining probability distributions over the elements of the graph. In this paper, we investigate the complexity of a class of composite graphs based on subdivision graphs and corona product evaluating the generalized graph entropies, and we present explicit formulae for the complexity of subdivision-corona type product graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of Boolean networks by using their matrices of support. We derive properties about the transients, fixed points, and limit cycles. We also study the graph properties of the phase spaces of Boolean networks, and show how to use matrices of support to construct Boolean networks with prescribed dynamics. [ABSTRACT FROM AUTHOR]

Graphs considered here are simple, finite and undirected. A graph is denoted by G and its vertex set by V (G) and edge set by E (G). Many researchers were attracted by two concepts introduced in [P. J. Slater, Domination and location in acyclic graphs, Networks17 (1987) 55–64; P. J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445–455; Y. Caro, A. Hansberg and M. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914]. One is locating domination and the other is fair domination. A subset D of V (G) is called a locating dominating set of G if for any u , v ∈ V − D , N (u) ∩ D ≠ N (v) ∩ D and both sets are non-empty. D is called a fair dominating set of G for any u , v ∈ D , | N (u) ∩ D | = | N (v) ∩ D | > 0. In this paper, both properties are combined and locating fair domination is studied. [ABSTRACT FROM AUTHOR]

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

Let G be a k -uniform hypergraph with vertex set V (G) and edge set E (G). A connected and acyclic hypergraph is called a supertree. For 0 ≤ α < 1 , the α -spectral radius of G is the largest H -eigenvalue of α D (G) + (1 − α) A (G) , where D (G) and A (G) are the diagonal tensor of the degrees and the adjacency tensor of G , respectively. In this paper, we determine the unique supertrees with maximum α -spectral radius among all k -uniform supertrees with m edges and independence number β for ⌈ m (k − 1) + 1 k ⌉ ≤ β ≤ m , among all k -uniform supertrees with given degree sequences, and among all k -uniform supertrees with m edges and matching number μ for 1 ≤ μ ≤ ⌊ m (k − 1) + 1 k ⌋ , respectively. [ABSTRACT FROM AUTHOR]

A dominating set D of a graph G = (V , E) is said to be certified, if every vertex v ∈ D has either 0 neighbors or at least two neighbors in V\D. The cardinality of a minimum certified dominating set of G is called the certified domination number of G and is denoted by γ cer (G). A graph G is said to be γ cer - E A stable if for any two vertices u , v ∈ V (G) such that u v ∉ E (G) , we have γ cer (G + u v) = γ cer (G). In this paper, we provide upper bounds for the γ cer -value of γ cer - E A stable graph. We also characterize the trees and unicyclic graphs that are γ cer - E A stable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the traces of the adjacency tensor of uniform hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and present some perturbation results for the traces of a hypergraph when its structure is locally changed. As an application we determine the unique hypertree with maximum Estrada index among all hypertrees with perfect matchings and given number of edges, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with girth 3 and given number of edges. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the vertex Padmakar–Ivan index of H -generalized join of graphs. As a consequence, we find the vertex Padmakar–Ivan index of the lexicographic product of graphs, the ideal-based zero-divisor graph and the zero-divisor graph of rings. In addition, we obtain the vertex Padmakar–Ivan index of the ideal-based zero-divisor graph and the zero-divisor graph of the ring of integers modulo n. [ABSTRACT FROM AUTHOR]

For a poset P = (X , ≤ P) , the strict-double-bound graph (sDB (P)) of P = (X , ≤ P) is the graph with the vertex set X such that u v ∈ E (sDB (P)) if and only if u ≠ v and there exist x ∈ X and y ∈ X distinct from u and v such that x ≤ P u ≤ P y and x ≤ P v ≤ P y. For a connected graph G , the strict-double-bound number ζ (G) is defined as min { l : there exists a poset P such that sDB(P) ≅ G ∪ N l } , where N l is the graph with l vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that ζ (P n × P m) ≤ 2 n + 2 m − 4 for n , m ≥ 2 , ζ (C 4 n × C 4 m) ≤ 8 n m + 4 for n , m ≥ 1 , and ζ (C n × C m) ≤ 4 n + 4 m − 4 for n , m ≥ 3. [ABSTRACT FROM AUTHOR]

The middle graph M (G) of a graph G is the graph obtained by subdividing each edge of G exactly once and joining all these newly introduced vertices of adjacent edges of G. It is known that the decision problems for Italian domination number γ I (G) , 2 -rainbow domination number γ r 2 (G) and Roman domination number γ R (G) are NP-complete. Recently, it was proved that γ I (M (G)) = γ r 2 (M (G)) = γ R (M (G)) = n for a graph G of order n. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs. [ABSTRACT FROM AUTHOR]

In the online facility assignment problem OFA (k , { c i } i = 1 k) , there exist k servers s 1 , ... , s k on a metric space where each s i has an integer capacity c i and a request arrives one-by-one. The task of an online algorithm is to irrevocably match a current request with one of the servers with vacancies before the next request arrives. As special cases for OFA (k , { c i } i = 1 k) , we consider OFA (k , { c i } i = 1 k) on a line , which is denoted by OFAL (k , { c i } i = 1 k) and OFAL eq (k , { c i } i = 1 k) , where the latter is the case of OFAL (k , { c i } i = 1 k) with equidistant servers. In this paper, we perform the competitive analysis for the above problems. As a natural generalization of the greedy algorithm grdy, we introduce a class of algorithms called MPFS (Most Preferred Free Servers) and show that any MPFS algorithm has the capacity-insensitive property, i.e., for any MPFS algorithm alg for OFA (k , { c i } i = 1 k) , if alg is c -competitive when c 1 = ⋯ = c k = 1 , then alg is c -competitive for general { c i } i = 1 k . By applying the capacity-insensitive property of the greedy algorithm grdy, we derive the matching upper and lower bounds 4 k − 5 on the competitive ratio of grdy for OFAL eq (k , { c i } i = 1 k). To investigate the capability of MPFS algorithms, we show that the competitive ratio of any MPFS algorithm alg for OFAL eq (k , { c i } i = 1 k) is at least 2 k − 1. Then, we propose a new MPFS algorithm idas (Interior Division for Adjacent Servers) for OFAL (k , { c i } i = 1 k) and show that the competitive ratio of idas for OFAL eq (k , { c i } i = 1 k) is at most 2 k − 1 , i.e., idas for OFAL eq (k , { c i } i = 1 k) is best possible in all the MPFS algorithms. We also give numerical experiments to investigate the performance of idas and grdy and show that idas performs better than grdy for distribution of request sequences with locality. [ABSTRACT FROM AUTHOR]

Secret-sharing is a vital topic of cryptography and has widely used in information protection. One technique to the development of secret-sharing schemes is primarily based on error-correcting codes. In this paper, we determine the minimal codewords of relative four-weight codes and have the utility in the secret sharing schemes. [ABSTRACT FROM AUTHOR]

In a graph G : = (V (G) , E (G)) , a subset M of the vertex set V (G) is a module (or interval, clan) of G if every vertex outside M is adjacent to all or none of M. The empty set, the singleton sets, and the full set of vertices are trivial modules. The graph G is indecomposable (or prime) if all its modules are trivial. If G is indecomposable, we say that an edge e of G is a removable edge if G − e is indecomposable (here G − e : = (V (G) , E (G) − { e })). The graph G is said to be unstable if it has no removable edges. For a positive integer k , the k th power G k of a graph G is the graph obtained from G by adding an edge between all pairs of vertices of G with distance at most k. A graph G of order n (i.e., | V (G) | = n) is said to be p -placeable into G k , if G k contains p edge-disjoint copies of G. In this paper, we answer a question, suggested by Boudabbous in a personal communication, concerning unstable graphs and packing into their fifth power as follows: First, we give a characterization of the unstable graphs which is deduced from the results given by Ehrenfeucht, Harju and Rozenberg (the theory of 2 -structures: a framework for decomposition and transformation of graphs). Second, we prove that every unstable graph G is 2 -placeable into G 5 . [ABSTRACT FROM AUTHOR]

A k - i n j e c t i v e - e d g e - c o l o r i n g o f a g r a p h G is a mapping f : E (G) → { 1 , 2 , ... , k } such that f (e 1) ≠ f (e 3) for any three consecutive edges e 1 , e 2 , e 3 of a path or a 3 -cycle. χ i ′ (G) = min { k | G has a k -injective-edge-coloring } is called the injective chromatic index of G. In this paper, we prove that for planar graphs G with Δ (G) ≥ 6 , (1) χ i ′ (G) ≤ 3 Δ (G) − 3 if g (G) ≥ 6 ; (2) χ i ′ (G) ≤ 3 Δ (G) − 4 if g (G) ≥ 7. [ABSTRACT FROM AUTHOR]

Nowadays, pairing-based cryptography researchers are looking for new parameters for standard security levels against the new number field sieve tower number field sieve algorithm. Recently, they have suggested new parameters for well-studied pairing-friendly curves with odd embedding degrees five and seven resistant to this attack. In this paper, we define optimal ate pairing on curves using sparse families with embedding degrees five and seven. We also provide details to perform the miller loop and the final exponentiation using addition chain process. Our theoretical results costs indicate that these families of curves offer the best performance in the computation of the optimal ate pairing at the 128-bit security level compared to Cocks–Pinch curves of embedding degrees five and seven. The improvement is about 8 , 8 % and 1 9 % faster than the optimal ate pairing previously computed on Cocks–Pinch curves of embedding degrees five and seven, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present two new graph structures that can be used to model information routing in heterogeneous communication networks. These are cardinality-constrained graphs (CCGs) and interval-constrained graphs (ICGs). In both cardinality- and interval-constrained graphs, the nodes in the graph are labeled. In a CCG, each label is associated with an upper bound. A path in a CCG is considered valid, if the number of nodes on the path with any particular label is at most the upper bound associated with that label. In an ICG, each label is associated with both an upper bound and a lower bound. A path in an ICG is considered valid, if the number of nodes on the path with any particular label is between the upper and lower bounds associated with that label. In this paper, we examine the computational complexities of finding valid paths in both CCGs and ICGs. In particular, we show that these problems are NP-hard. Additionally, we propose novel algorithms for finding paths in CCGs and a subset of ICGs. We also demonstrate the effectiveness of our algorithms on CCGs and ICGs of various sizes. The past decade has seen a surge in the degree of hardware-heterogeneous device interconnectivity, Additionally, there has been a proliferation of resource-constrained wireless sensor networks. Thus, the problems of finding valid paths in CCGs and ICGs are of interest. Nodes in wireless sensor networks can be faulty, vulnerable, energy-constrained, or untrusted. These nodes may also have differing degrees of performance or power efficiency. This can pose a significant challenge in routing the flow of information between nodes. The heterogeneity of nodes in such networks is captured through the use of CCGs and ICGs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a variant of strong metric dimension, called the fault-tolerant strong metric dimension. A strong resolving set S for G is fault-tolerant if S \ { s } is also a strong resolving set, for each s in S , and the fault-tolerant strong metric dimension of G is the minimum cardinality of such a set and is denoted by dim f s (G). [ABSTRACT FROM AUTHOR]

In this paper, we consider Tower of Hanoi graphs and study their matching properties. Explicit system of recurrences is derived for the matching polynomials of these graphs and their appropriate truncated variants. Consequently, we obtain exact formula for the numbers of maximum matchings in Hanoi graphs using matching polynomials, which is a new approach for old one problem. [ABSTRACT FROM AUTHOR]

The eccentric graph of a graph G , denoted by G e , is a derived graph with the vertex set same as that of G and two vertices in G e are adjacent if one of them is an eccentric vertex of the other. The process of constructing iterative eccentric graphs, denoted by G e k is called eccentrication. A graph G is said to be ℰ − cyclic (t , l) if G , G e , G e 2 , ... , G e k , G e k + 1 , ... , G e k + l are the only non-isomorphic graphs, and the graph G e k + l + 1 is isomorphic to G e k . In this paper, we prove the existence of an ℰ -cycle for any simple graph. The importance of this result lies in the fact that the enumeration of eccentrication of a graph reduces to a finite problem. Furthermore, the enumeration of a corresponding sequence of graph parameters such as chromatic number, domination number, independence number, minimum and maximum degree, etc., reduces to a finite problem. [ABSTRACT FROM AUTHOR]

The k -partition dimension of graphs was introduced as generalization of partition dimension of graphs by Estrado–Moreno in 2020. Its applications can be seen in several areas including robot navigation, network designing, image processing and chemistry. Rotationally symmetric graphs are important in computer network topologies due to uniform rate of data transformation to all nodes. In this paper, we compute the lower bound of the 2 -partition dimension of r -regular graphs for r ≥ 3 and also calculate the 2 -partition dimension of the families of rotationally symmetric graphs, namely, prism graph and n -sunlet graph. [ABSTRACT FROM AUTHOR]

Separating codes have been studied due to their applications to digital finger printing, the state assignments, automata theory and to construct hash functions. In this paper, we study the necessary and sufficient conditions for a code to be a (3 , 1) and (3 , 2) -separating systems for q-ary level and also satisfy its intersecting properties. We also construct a bound for (3 , 1) separating system. [ABSTRACT FROM AUTHOR]

The basic intention of this paper is to impart the importance of Nano topology induced by graph. Nano topology induced by graph paves us a method to find the well-known parameter of a graph, namely its domination number as well as all the possible dominating sets of a graph. In general, each paper, depending upon the author's axe (or the pen), has its own point of emphasis. Accordingly, in this paper, the computational and algorithmic aspects of graph are emphasized. By experience, we know that to solve a problem using graph theory leads to a large graph virtually impossible to analyze without the aid of a computer. Hence, to simplify the problem we have used a Java program code that suits even for all graphs of finite number of vertices. We have made an attempt to find the proper location of guards and minimize the number of guards to observe all the warning lights in a nuclear power plant using such a Java program code. [ABSTRACT FROM AUTHOR]

A dominating set D ⊂ V (G) is said to be a captive dominating set of G if 〈 D 〉 has no isolated vertex (D is a total dominating set) and each vertex v ∈ D is adjacent to at least one vertex in V − D. A captive dominating set D is said to be a minimal captive dominating set if no proper subset D ′ of D is a captive dominating set. The minimum cardinality of a minimal captive dominating set of G is called the captive domination number of G which is denoted by γ c a (G). In this paper, we have characterized some results, determine the values of the domination-related parameters for the graph and its splitting graph, relate captive domination and packing number of a graph, etc. We have also constructed graphs for which γ c a > γ t. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple graph. A dominating set of G is a subset S ⊆ V such that every vertex not in S is adjacent to at least one vertex in S. The cardinality of a smallest dominating set of G , denoted by γ (G) , is the domination number of G. A dominating set S is called a super dominating set of G , if for every vertex u ∈ S ¯ = V − S , there exists v ∈ S such that N (v) ∩ S ¯ = { u }. The cardinality of a smallest super dominating set of G , denoted by γ sp (G) , is the super domination number of G. In this paper, we study super domination number of some graph classes and present sharp bounds for some graph operations. [ABSTRACT FROM AUTHOR]

Given a graph G = (V (G) , E (G)) and a proper total k -coloring μ : V (G) ∪ E (G) → { 1 , 2 , ... , k } , we call μ neighbor sum distinguishing total coloring provided h (v) ≠ h (u) for any u v ∈ E (G) where h (v) = μ (v) + ∑ u v ∈ E (G) μ (u v) for any v ∈ V (G). Neighbor sum distinguishing total coloring was first defined by Pilśniak and Woźniak. They conjectured Δ (G) + 3 colors enable any graph G to admit such a coloring. The neighbor sum distinguishing total chromatic number χ Σ ′ ′ is the minimum integer where a graph is needed for this coloring. In this paper, we present two conclusions that χ Σ ′ ′ ≤ Δ (G) + 2 provided there are no 3-cycles adjacent to 4-cycles in a planar graph G with Δ (G) ≥ 8 without cut edges, and χ Σ ′ ′ ≤ Δ (G) + 3 provided there are no 4-cycles intersecting with 6-cycles in a planar graph G with Δ (G) ≥ 7. [ABSTRACT FROM AUTHOR]

An Italian dominating function of a graph G = (V , E) with vertex set V is defined as a function f : V → { 0 , 1 , 2 } , which satisfies the condition that for every v ∈ V with f (v) = 0 , ∑ u ∈ N (v) f (u) ≥ 2. The weight of an Italian dominating function on G is the sum f (V) = ∑ u ∈ V f (v) and the Italian dominating number, and γ I (G) indicates the minimum weight of an Italian dominating function f. In this paper, the structure of the generalized Sierpiński networks is investigated using the bounds of Italian domination number of graphs and the methods of mathematical induction and reduction to absurdity. Then, the Italian domination on the generalized Sierpiński networks S (G , t) is obtained, where G denotes any special class of Path P n , Cycle C n , Wheel W n , Star K 1 , n , and complete bipartite graph K m , n . [ABSTRACT FROM AUTHOR]

A signed network Σ = (G , σ) with the underlying graph G = (V , E) , used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or − 1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones. [ABSTRACT FROM AUTHOR]

An injective k -edge-coloring of a graph G = (V , E) is an assignment ω : E → { 1 , 2 , ... , k } of colors to the edges of G such that any two edges e and f receive distinct colors if there exists an edge g = x y different from e and f such that e is incident on x and f is incident on y. The least number of colors required by any injective edge coloring of G is called the injective chromatic index of G and is denoted by χ i ′ (G). In this paper, we give tight upper bounds of the injective chromatic index of various standard graph products and operations, including the Cartesian product, lexicographic product, corona product, edge corona product, join, subdivision and Mycielskian of a graph. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph where V is the set of vertices of G and E is the set of edges of G. The distance from the vertex w ∈ V to the edge  e = u v ∈ E is given by d (e , w) = min { d (u , w) , d (v , w) }. A subset S e ⊂ V is called an edge metric generator for G if for every two distinct edges e 1 , e 2 ∈ E , there exists a vertex w ∈ S such that d (e 1 , w) ≠ d (e 2 , w). The edge metric generator with the minimum number of vertices is called an edge metric basis for G and the cardinality of the edge metric basis is called the edge metric dimension denoted by dim e (G). A subset S m ⊂ V is called a mixed metric generator for G if for every two distinct elements x , y ∈ V ∪ E , there exists a vertex w ∈ S such that d (x , w) ≠ d (y , w). A mixed metric generator containing a minimum number of vertices is called a mixed metric basis for G and the cardinality of a mixed metric basis is called the mixed metric dimension denoted by dim m (G). In this paper, we study the edge metric dimension and the mixed metric dimension of a plane graph  T n . [ABSTRACT FROM AUTHOR]

A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } satisfying (i) if f (v) = 0 then there must be at least two neighbors assigned two under f or one neighbor w with f (w) = 3 ; and (ii) if f (v) = 1 then v must be adjacent to a vertex w such that f (w) ≥ 2. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on G if the set of vertices labeled 0 induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number γ tdR oi (G) is the minimum weight of an OITDRDF on G. In this paper, we establish various bounds on γ tdR oi (G). In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results. [ABSTRACT FROM AUTHOR]

In this paper, we determine all simple block-transitive 3-designs on 65 points with block stabilizer of order greater than 2, admitting the 2-transitive action of the group S z (8). As a result, we find 23 3-designs on 65 points that have S z (8) as an automorphism group. Also we construct four 3-(3 6 , 1 8 , λ 1 ) designs for λ 1 ∈ { 7 6 8 , 5 1 8 4 , 6 9 1 2 , 9 2 1 6 } and two 3-(3 6 , 1 7 , λ 2 ) designs for λ 2 ∈ { 2 8 8 0 , 8 6 4 0 }. All of these designs have S 6 (2) as the full automorphism group. Some of these designs are flag and anti-flag-transitive. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Z (R) be its zero-divisors set. The zero-divisor graph of R , denoted by Γ (R) , is an undirected graph with vertex set Z (R) ∗ = Z (R) ∖ { 0 } and two distinct vertices a and b are adjacent if and only if a b = 0. In this paper, for n = p R q S where p and q are primes (p < q) and R and S are positive integers, we calculate the A α spectrum of the graphs Γ (ℤ n). [ABSTRACT FROM AUTHOR]

The principal left ideals of a regular semigroup have been abstractly described by Nambooripad as normal categories. A semigroup S is said to be a band if a 2 = a for all a ∈ S. Since bands are regular semigroups, the category of principal left ideals is a normal category. This paper characterizes the normal categories which arise as categories of principal left ideals of bands. [ABSTRACT FROM AUTHOR]

All the graphs considered in this paper are simple and undirected. Let G = (V (G),E(G)) be a simple undirected graph. A function f : V (G) →{0, 1, 2} is called root cube mean cordial labeling if the induced function f∗ : E(G) →{0, 1, 2} defined by f∗(uv) = ⌊(f(u))3 +(f(v))3 2 ⌋ satisfies the condition |vf(i) − vf(j)|≤ 1 and |ef(i) − ef(j)|≤ 1 for any i,j ∈{0, 1, 2}, where vf(x) and ef(x) denotes the number of vertices and number of edges with label x, respectively, and ⌊x⌋ denotes the greatest integer less than or equals to x. The G is called root cube mean cordial if it admits root cube mean cordial labeling. In this paper, we have discussed root cube mean cordial labeling of some graphs. Also, we have provided some graphs which are not root cube mean cordial. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]