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This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs. [ABSTRACT FROM AUTHOR]

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has recently been introduced by Park and Ihm.15 In this paper, we examine properties of strong matching preclusion for alternating group graphs, by finding their strong matching preclusion numbers and categorizing all optimal solutions. More importantly, we prove a general result on taking a Cartesian product of a graph with K2 (an edge) to obtain the corresponding results for split-stars. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple, connected graph of order n and size m. Then, G = (V , E) is said to be edge (m , k) -choosable, if there exists a collection of subsets of the edge set, { S k (e) : e ∈ E (G) } of cardinality k , such that S k (e 1) ∩ S k (e 2) = ∅ , whenever e 1 and e 2 are incident. This paper initiates a study on edge (m , k) -choosability of certain fundamental classes of graphs and determines the maximum value of k , for which the given graph G is edge (m , k) -choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph. [ABSTRACT FROM AUTHOR]

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path x y z , if there are pegs in x and y and exists a hole in z , then x can jump over y into z , and after that, the peg in y is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex s , starting with a hole s , a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs. [ABSTRACT FROM AUTHOR]

In the last two years, many chaotic or hyperchaotic systems with megastability have been reported in the literature. The reported systems with megastability are mostly developed from their dynamic equations without any reference to the physical systems. In this paper, the dynamics of a single-link manipulator is considered to observe the existence of interesting dynamical behaviors. When the considered dynamical system is excited with (a) periodically forced input or (b) quasi-periodically forced input, it indicates the existence of megastability. This paper reports megastability in a physical dynamical system with infinitely many equilibria. The considered system has other dynamical behaviors like chaotic, quasi-periodic and periodic. These behaviors are analyzed using Lyapunov spectrum, bifurcation diagram and phase plots. The simulation results reveal that the objectives of the paper are achieved successfully. [ABSTRACT FROM AUTHOR]

Super strongly perfect graphs and their association with certain other classes of graphs are discussed in this paper. It is observed that every split graph is super strongly perfect. An existing result on super strongly perfect graphs is disproved providing a counter example. It is also established that if a graph G contains a cycle of odd length, then its line graph L(G) is not always super strongly perfect. Complements of cycles of length six or above are proved to be non-super strongly perfect. If a graph is strongly perfect, then it is shown that they are super strongly perfect and hence all P 4 -free graphs are super strongly perfect. [ABSTRACT FROM AUTHOR]

A signed graph Σ is a simple undirected graph in which each edge is either positive or negative. Restrained dominating set D in Σ is a restrained dominating set of the underlying graph | Σ | where the subgraph induced by the edges across Σ [ D : V ∖ D ] and within V ∖ D is balanced. The minimum cardinality of a restrained dominating set of Σ is called the restrained domination number, denoted by γ r (Σ). In this paper, we initiate the study on various critical concepts to investigate the effect of edge removal or edge addition on restrained domination number in signed graphs. [ABSTRACT FROM AUTHOR]

The domatic number d (G) of a graph G is the maximum number of disjoint dominating sets in a dominating set partition of a graph G. For any graph G , d (G) ≤ δ (G) + 1 , where δ (G) is the minimum degree of G , and G is domatically full if the equality holds, i.e., d (G) = δ (G) + 1. In this paper, we characterize domatically full Cartesian products of a path of order 2 and a tree of order at least 3. Moreover, we show a characterization of the Cartesian product of a longer path and a tree of order at least 3. By using these results, we also show that for any two trees of order at least 3, the Cartesian product of them is domatically full. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new kind of graph related to a unitary module M on a commutative ring R with identity, namely the annihilators comaximal graph of submodules of M , denoted by G ∗ (M). The (undirected) graph G ∗ (M) is with vertices of all non-trivial submodules of M and two vertices N , K of G ∗ (M) are adjacent if and only if their annihilators are comaximal ideals of R , i.e. Ann (N) + Ann (K) = R. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of G ∗ (M) and the module-theoretic properties of M. We study the annihilators comaximal graph G ∗ (ℤ n) in terms of the powers of the decomposition of n to product distinct prime numbers in some special cases. [ABSTRACT FROM AUTHOR]

In recent years, the study of social-based recommender systems has become an active research topic. We incorporate a combination of static and dynamic interest characteristics to predict users' real-time dynamic interests, which has rarely been considered in previous studies. In this paper, we propose a graph neural network social recommendation model that integrates static and dynamic feature relationships (FSDFR-GNNSR). The model uses a graph embedding algorithm to extract static features of users and movies, and takes the static features as input to gated recurrent unit (GRU), so that the model can take static features into consideration while modeling user dynamic behavior. Finally, we use graph attention networks to represent the dynamic influence of friends, simplify the update strategy of second-order neighbor nodes. We apply graph pooling operations to improve the generalization ability of the algorithm. Empirical analyses on real datasets show that the proposed approach achieves superior performance to existing approaches. [ABSTRACT FROM AUTHOR]

The concept of Sombor indices (SO) of a graph was recently introduced by Gutman and the reduced Sombor index SO red of a graph G is defined by SO red (G) = ∑ u v ∈ E (G) (d G (u) − 1) 2 + (d G (v) 2 − 1) , where d G (v) is the degree of the vertex v. In this paper, we study the extremal properties of the reduced Sombor index and characterize the bicyclic graphs with extremal SO red -value. [ABSTRACT FROM AUTHOR]

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let G = Z 2 k − 1 × Z 4 be a finite abelian 2-group of order 2 k + 1 where k ≥ 1. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on k. [ABSTRACT FROM AUTHOR]

The sum-connectivity index of a graph G is defined as the sum of weights 1 / d (u) + d (v) over all edges u v of G , where d (u) and d (v) are the degrees of the vertices u and v in G , respectively. A graph G is called quasi-tree, if there exists u ∈ V (G) such that G − u is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs. [ABSTRACT FROM AUTHOR]

In this paper, we characterize uniquely colorable graphs with equal chromatic and game chromatic numbers. [ABSTRACT FROM AUTHOR]

We address in this paper the connected p -median problem on complete multi-layered graphs. We first solve the corresponding 1-median problem on the underlying graph in linear time based on the structure of the induced path. For p ≥ 2 , we prove that the connected p -median contains at least one vertex in the layered set corresponding to the 1-median or its adjacent vertices in the induced path. Then, we develop an O (n log n) algorithm that solves the problem on a complete multi-layered graph with n vertices. [ABSTRACT FROM AUTHOR]

The forgotten topological index of a graph , denoted by F () , is defined as the sum of weights d (u) 2 + d (v) 2 overall edges u v of , where d (u) denotes the degree of a vertex u. The graph is called a quasi-unicyclic graph if there exists a vertex v ∈ () such that − v is a connected graph with a unique cycle. In this paper, we give sharp upper and lower bounds for the F-index (forgotten topological index) of the quasi-unicyclic graphs. [ABSTRACT FROM AUTHOR]

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new graph product, namely, bowtie product. We determine the generalized characteristic polynomial of the bowtie product of graphs and consequently, we obtain the characteristic polynomial of the adjacency martix, the Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix of this graph. Also, from these results, we obtain the L -spectra of some families of bowtie product of graphs. As an application, we obtain infinite pairs of L -cospectral graphs and construct infinite number of L -integral graphs. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a graph G is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored P 4 in G. In this paper, we show that the Cartesian product of any two cycles, except C 3 □ C 3 and C 3 □ C 5 , has a 5 -star coloring. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for proving plane geometry theorems stated by text and diagram in a complementary way. The problem of proving plane geometry theorems involves two challenging subtasks, being theorem understanding and theorem proving. This paper proposes to consider theorem understanding as a problem of extracting relations from text and diagram. A syntax–semantics (S2) model method is proposed to extract the geometric relations from theorem text, and a diagram mining method is proposed to extract geometry relations from diagram. Then, a procedure is developed to obtain a set of relations that is consistent with the given theorem with high confidence. Finally, theorem proving is conducted by using the existing proving methods which take the extracted geometric relations as input. The experimental results show that the proposed theorem proving algorithm can prove 86% of plane geometry theorems in the test dataset of 200 theorems, which is all the theorems in the popular textbook. The proposed algorithm outperforms the existing algorithms mainly because it can extract relations not only from text but also from diagram. [ABSTRACT FROM AUTHOR]

The Estrada index is a graph-spectrum-based invariant having an important role in chemistry and physics. In this paper, we define the generalized adjacency Estrada index of a graph and obtain some lower and upper bounds for the generalized adjacency Estrada index in terms of various graph parameters associated with the structure of a graph. Further, we characterize the extremal graphs attaining these bounds. We also highlight the relationship between generalized adjacency Estrada index and generalized adjacency energy. Using these results, we obtain the improved bounds for the Estrada index based on the adjacency eigenvalues of a graph. [ABSTRACT FROM AUTHOR]

The derived graph of a simple graph G , denoted by [ G ] † , is the graph having the same vertex set as G , in which two vertices are adjacent if and only if their distance in G is two. The spectrum of derived graph of G is called the second-stage spectrum of G. In this paper, we will determine the second-stage spectrum (Laplace and signless Laplace spectrum) of the corona of two regular graphs with diameter less than or equal to two. The energy of the derived graph is called the second-stage energy of G. Here, we proved that the class of graphs [ K n 1 ∘ K n 2 ] † is integral if and only if n 2 is a perfect square and the second-stage energy depends only on number of vertices. Moreover, we discuss some applications like the number of spanning trees, the Kirchhoff index and Laplacian-energy-like invariant of the newly constructed graph. [ABSTRACT FROM AUTHOR]

Let G be a graph and f = (V 0 , V 1 , V 2) be a Roman dominating function defined on G. Let (v 1 , v 2 , ... , v k) be some ordering of the vertices of V 2 . For any x ∈ V 0 , code (x) is defined by code (x) = (d (x , v 1) , ... , d (x , v k)). If for all x , y ∈ V 0 , x ≠ y , we have code (x) ≠ code (y) , that is d (x , v) ≠ d (y , v) , for some v ∈ V 2 , then f is called a resolving Roman dominating function (RDF) on G. The weight of a resolving RDF f = (V 0 , V 1 , V 2) on G is w (f) = | V 1 | + 2 | V 2 |. The minimum weight of a resolving RDF on G is called the resolving Roman domination number of G and is denoted by γ R R (G). A resolving RDF on G with weight γ R R (G) is called a γ R R -function on G. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order. [ABSTRACT FROM AUTHOR]

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation A * C B or an HNN-extension * ϕ C , where C is a finite group and [ A : C ] = [ B : C ] = 2 and ϕ ∈ Aut (C). In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and d i is the degree of its i th vertex ( d i is the degree of v i ), then the Zagreb matrix of G is the square matrix of order n whose (i , j) entry is equal to d i + d j if the i th and j th vertex of G are adjacent, and zero otherwise. The Zagreb energy Z E (G) of a graph G is defined as the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we compute the Zagreb energy for some of the specific graphs, edge deleted graphs and complements graphs. Moreover, some properties of the eigenvalues and bounds for Zagreb energy are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we give a procedure to construct quantum circuits for implementing discrete-time quantum walks on a family of Cayley graphs called cubelike graphs. We construct these circuits on IBM's Qiskit platform and demonstrate an implementation of the quantum hitting times on cubelike graphs. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. This conjecture, if proved, will generalize the result of hitting times of discrete-time quantum walks on hypercubes to general family of cubelike graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n and d i be the degree of the vertex v i , for i = 1 , 2 , ... , n. The ISI matrix of G is the square matrix of order n whose (i , j) -entry is equal to d i d j d i + d j if v i is adjacent to v j , and zero otherwise. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n , be the eigenvalues of ISI matrix. The ISI energy of a graph G , denoted by ξ ISI (G) , is defined as the sum of the absolute values of the eigenvalues of ISI matrix. In this paper, we prove that the star has the minimum ISI energy among trees. [ABSTRACT FROM AUTHOR]

In this paper, we present the weighted iterated friendship graphs and study the trapping problem on the weighted iterated friendship graphs. It can be found that for 0 < r < 1 and r = 1 , the relationship between the average trapping time (ATT) and network size is sublinear and linear, respectively. By controlling the parameters of the weighted iterated friendship graphs, the models are changed to the self-similar weighted networks. The average shortest weighted path (ASWP) in the self-similar weighted friendship graphs is studied. The results show that when 0 < r < 1 , the ASWP is bounded, and when r = 1 , the ASWP is linearly related to the order of the networks. [ABSTRACT FROM AUTHOR]

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. [ABSTRACT FROM AUTHOR]

The non-commuting graph associated to a group has non-central elements of the graph as vertices and two elements x and y do not form an edge if and only if x y = y x. In this paper, we consider non-commuting graphs associated to dihedral and semidihedral groups. We investigate their metric properties such as center, periphery, eccentric graph, closure and interior. We also perform various types of metric identifications on these graphs. Moreover, we generate metric and metric-degree polynomials of these graphs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a guideline for plotting the bifurcation diagrams of chaotic systems. We discuss numerical and mathematical facts in order to obtain more accurate and more elegant bifurcation diagrams. The importance of transient time and the phenomena of critical slowing down are investigated. Some critical issues related to multistability are discussed. Finally, a solution for fast obtaining an accurate sketch of the bifurcation diagram is presented. The solution is based on running the system for only one sample in each parameter value and using the system's state in the previous value of the parameter as the initial condition. [ABSTRACT FROM AUTHOR]

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are "writhe-like" invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link. [ABSTRACT FROM AUTHOR]

The way an author or a group of authors are cited tells more about the real impact of their work than authorship and collaborations. Indeed, the connections within the scientific community can be more accurately elicited from the co-citation graph than from the collaboration graph. We suggest some indices that can be drawn from the co-citation graph in order to capture the relevance of individual authors and the relevance of groups of authors. [ABSTRACT FROM AUTHOR]

This paper presents a four-dimensional hyperchaos Qi system, obtained by adding linear term and nonlinear term of nonlinear controller to Qi chaos system. The hyperchaos Qi system is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices, m edges having Laplacian eigenvalues μ 1 , μ 2 , ... , μ n − 1 , μ n = 0. The Laplacian energy LE (G) is defined as LE (G) = ∑ i = 1 n | μ i − d ¯ | , where d ¯ = 2 m n is the average degree of G. Radenković and Gutman conjectured that among all trees of order n , the path graph P n has the smallest Laplacian energy. Let n (d) be the family of trees of order n having diameter d. In this paper, we show that Laplacian energy of any tree T ∈ n (4) is greater than the Laplacian energy of P n , thereby proving the conjecture for all trees of diameter 4. We also show the truth of conjecture for all trees with number of non-pendent vertices at most 9 n 2 5 − 2. Further, we give some sufficient conditions for the conjecture to hold for a tree of order n. [ABSTRACT FROM AUTHOR]

Let L be a lattice, I an ideal of L and T (I) the set of all elements of L which are not prime to I. In this paper, we investigate some interesting properties of T (I) and introduce the total graph of a lattice L with respect to an ideal I. It is the graph with all elements of L as vertices and for distinct a , b ∈ L , the vertices a and b are adjacent if and only if a ∨ b ∈ T (I). We investigated the basic properties of this graph and its induced subgraphs as diameter, girth, clique number, cut vertex and independence number. [ABSTRACT FROM AUTHOR]

Let f : V → { 0 , 1 , 2 } be a function on a graph G = (V , E). A vertex v with f (v) = 0 is said to be undefended with respect to f if it is not adjacent to a vertex u with f (u) > 0. A function f is called a weak Roman dominating function (WRDF) if each vertex v with f (v) = 0 is adjacent to a vertex u with f (u) > 0 , such that the function f ′ defined by f ′ (v) = 1 , f ′ (u) = f (u) − 1 and f ′ (w) = f (w) for all w ∈ V ∖ { v , u } , has no undefended vertex. The weight of a WRDF is the sum of its function values over all vertices, and the weak Roman domination number γ r (G) is the minimum weight of a WRDF in G. In this paper, we consider the effects of edge deletion on the weak Roman domination number of a graph. We show that the deletion of an edge of G can increase the weak Roman domination number by at most 1. Then we give a necessary condition for γ r -ER-critical graphs, that is, graphs G whose weak Roman domination number increases by the deletion of any edge. Restricted to the class of trees, we provide a constructive characterization of all γ r -ER-critical trees. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph. A function f : V → { 0 , 1 , 2 } is called a Roman dominating function if every vertex v with f (v) = 0 is adjacent to a vertex u with f (u) = 2. If further the set V 0 = { v : f (v) = 0 } is an independent set, then f is called an outer independent Roman dominating function (OIRDF). Let w (f) = ∑ v ∈ V f (v) and γ oiR (G) = min { w (f) : f  is an OIRDF of  G }. Then γ o i R (G) is called the outer independent Roman domination number of G. In this paper, we prove that the decision problem for γ o i R (G) is NP-complete for chordal graphs. We also show that γ o i R (G) is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, n ≥ 2 be a positive integer and M n (R) be the set of all n × n matrices over R. For a matrix A ∈ M n (R) , Tr (A) is the trace of A. The trace graph of the matrix ring M n (R) , denoted by Γ t (M n (R)) , is the simple undirected graph with vertex set { A ∈ M n (R) ∗ : there exists  B ∈ M n (R) ∗  such that Tr (A B) = 0 } and two distinct vertices A and B are adjacent if and only if Tr (A B) = 0. The ideal-based trace graph of the matrix ring M n (R) with respect to an ideal I of R , denoted by Γ I t (M n (R)) , is the simple undirected graph with vertex set M n (R) ∖ M n (I) and two distinct vertices A and B are adjacent if and only if Tr (A B) ∈ I. In this paper, we investigate some properties and structure of Γ I t (M n (R)). Further, it is proved that both Γ t (M n (R)) and Γ I t (M n (R)) are Hamiltonian. [ABSTRACT FROM AUTHOR]

A matrix with entries 0 , 1 is graphical if it is symmetric and all its diagonal entries are zero. Let G 1 , G 2 and G 3 be graphs defined on the same set of vertices. The graph G 3 is said to be the matrix product of graphs G 1 and G 2 , if A (G 1) A (G 2) = A (G 3) , where A (G i) is the adjacency matrix of the graph G i , 1 ≤ i ≤ 3. In such a case, we say that G 1 and G 2 are companions of each other. The main purpose of this paper is to design an algorithm to check whether a given graph G has a companion. We derive conditions on n and m so that the generalized wheel graph, denoted by W m , n , has a companion and also show that the m th power of the path graph P n has no companion. Finally, we indicate a possible application of the algorithm in a problem of coloring of edges of the complete graph K n . [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n and let RD (G) be the reciprocal distance matrix (also called Harary matrix) of the graph G. Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n be the eigenvalues of the reciprocal distance matrix RD (G) of the connected graph G called the reciprocal distance eigenvalues of G. The Harary energy HE (G) of a connected graph G is defined as sum of the absolute values of the reciprocal distance eigenvalues of G , that is, HE (G) = ∑ i = 1 n | ρ i |. In this paper, we establish some new lower and upper bounds for HE (G) , in terms of different graph parameters associated with the structure of the graph G. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of k largest adjacency eigenvalues of a connected graph. [ABSTRACT FROM AUTHOR]

For any integer k ≥ 1 , a minus total k -dominating function is a function f : V (G) → { − 1 , 0 , 1 } satisfying ∑ w ∈ N (v) f (w) ≥ k for every v ∈ V (G) , where N (v) = { u ∈ V (G) | u v ∈ E (G) }. The minimum of the values of ∑ v ∈ V (G) f (v) , taken over all minus total k -dominating functions f , is called the minus total k -domination number and is denoted by γ t k − (G). In this paper, we initiate the study of minus total k -domination in graphs, and we present different sharp bounds on γ t k − (G). In addition, we determine the minus total k -domination number of some classes of graphs. Some of our results are extensions of known properties of the minus total domination number γ t − (G) = γ t 1 − (G). [ABSTRACT FROM AUTHOR]

The irregularity of a simple undirected graph G is defined as irr (G) = ∑ u v ∈ E (G) | d G (u) − d G (v) | , where d G (u) denotes the degree of a vertex u ∈ V (G). The total irregularity of a graph as a new measure of graph irregularity is defined as irr t (G) = 1 2 ∑ u , v ∈ V (G) | d G (u) − d G (v) |. In this paper, the irregularity and the total irregularity of fractal graphs and the derived graphs from a class of fractal graphs are investigated. Exact formulae are presented for the computation of the irregularity and total irregularity of fractal-type graphs in terms of the parameters of the underlying graphs. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a double Roman dominating function is a function f : V (G) → { 0 , 1 , 2 , 3 } having the property that if f (v) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f (w) = 3 , and if f (v) = 1 , then vertex v must have at least one neighbor with f (w) ≥ 2. The weight of a double Roman dominating function f is the value w (f) = ∑ v ∈ V (G) f (v). The double Roman domination number of a graph G , denoted by γ d R (G) , equals the minimum weight of a double Roman dominating function on G. The double Roman domination subdivision number sd γ d R (G) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd γ d R (G) is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs. [ABSTRACT FROM AUTHOR]

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that | c (N (v)) | ≥ min { r , d (v) } for each vertex v ∈ V (G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of brick product graphs. [ABSTRACT FROM AUTHOR]

In this paper, the commuting graphs associated with finite groups are discussed. As the main theme, the explicit formulas for the number of spanning trees of commuting graphs associated with some specific groups, such as the Suzuki simple groups, the projective special linear groups, and some certain AC-groups, are obtained. [ABSTRACT FROM AUTHOR]

Decision diagrams are a data structure suitable for reversible circuit synthesis. Functional decision diagrams (FDDs) are particularly convenient in synthesis with Toffoli gates, since the functional expressions for decomposition rules used in them are similar to the functional expressions of Toffoli gates. The main drawback of reversible circuit synthesis based on decision diagrams is the usually large number of ancilla lines. This paper presents two methods for the reduction of the number of ancilla lines in reversible circuits derived from FDDs by selecting the order of implementation of nodes. In the first method, nodes are implemented by levels, starting from the bottom level to the top. The method uses appropriately defined level dependency matrices for choosing the optimal order of implementation of nodes at the same level. In this way, the optimization is performed level by level. The second method uses a diagram dependency matrix expressing mutual dependencies among all the nodes in the diagram. This method is computationally more demanding than the first method, but the reductions of both the number of lines and the Quantum cost of the circuits are larger. [ABSTRACT FROM AUTHOR]

If R is a ring then the square element graph q (R) is the simple undirected graph whose vertex set consists of all non-zero elements of R and two distinct vertices u , v are adjacent if and only if u + v = x 2 for some x ∈ R ∖ { 0 }. In this paper, we provide some necessary and sufficient conditions for the connectedness of q (R) , where R is a ring with identity. We mainly characterize some special class of ring R which we call square-subtract ring for which the graph q (M n (R)) is connected. [ABSTRACT FROM AUTHOR]

For a set W of vertices and a vertex v in a graph G , the k -vector r 2 (v | W) = (a G (v , w 1) , ... , a G (v , w k)) is the adjacency representation of v with respect to W , where W = { w 1 , ... , w k } and a G (x , y) is the minimum of 2 and the distance between the vertices x and y. The set W is an adjacency resolving set for G if distinct vertices of G have distinct adjacency representations with respect to W. The minimum cardinality of an adjacency resolving set for G is its adjacency dimension. It is clear that the adjacency dimension of an n -vertex graph G is between 1 and n − 1. The graphs with adjacency dimension 1 and n − 1 are known. All graphs with adjacency dimension 2 , and all n -vertex graphs with adjacency dimension n − 2 are studied in this paper. In terms of the diameter and order of G , a sharp upper bound is found for adjacency dimension of G. Also, a sharp lower bound for adjacency dimension of G is obtained in terms of order of G. Using these two bounds, all graphs with adjacency dimension 2, and all n -vertex graphs with adjacency dimension n − 2 are characterized. [ABSTRACT FROM AUTHOR]

A Latin square graph Γ (L) is a simple graph associated with a Latin square L. In this paper, we consider a Latin square graph Γ (L) in which n ≥ 6 and improve the upper bound of the domination number γ (Γ (L)) by showing that γ (Γ (L)) ≤ n − 2. Moreover, we study certain domination numbers like medium domination, accurate domination, independent transversal domination and vertex covering transversal domination for Latin square graphs of order n to give useful facts. [ABSTRACT FROM AUTHOR]