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A 2-distance coloring of a graph is a proper k-coloring in which any two vertices with distance at most two get different colors. The 2-distance number is the smallest number k such that G has a 2-distance k-coloring, denoted as χ2(G). In 1977, Wegner conjectured that for each planar graph G with maximum degree Δ, χ2(G) ≤ 7 if Δ ≤ 3, χ2(G) ≤ Δ + 5 if 4 ≤ Δ ≤ 7, and χ2(G) ≤⌊3Δ 2 ⌋ + 1 if Δ ≥ 8. In 2001, Thomassen supported the conjecture by proving the case Δ = 3. The conjecture is still open even for Δ = 4. In this paper, we show that χ2(G) ≤ 17 for the case Δ ≤ 5 which improves the upper bound 18 recently obtained by Hou et al. [ABSTRACT FROM AUTHOR]

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by d) and calculate the maximum diameter that is denoted by f(t,d) after deleting t edges from hypercube graphs Qn for some n. [ABSTRACT FROM AUTHOR]

The edge incident 2-edge coloring of a graph G is an edge coloring of the graph G such that not more than two colors are assigned to the edges incident to an edge e = uv in G. In other words, for every edge e in G, the edge e and all the edges that are incident to the edge e is in at most two different color classes. The edge incident 2-edge coloring number ψein2′(G) is the maximum number of colors in any edge incident 2-edge coloring of G. The main objective of this paper is to study the edge incident 2-edge coloring concept and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds. [ABSTRACT FROM AUTHOR]

Let A (G) and D (G) be the adjacency matrix and the diagonal matrix of vertex degrees of a graph G , respectively. For any real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). Let G ¯ be complement of G. Define S α (G) = A α (G ¯) − A α (G) = (1 − α) J + (α n − 1) I − 2 A α (G) , α ∈ [ 0 , 1 ] , where I and J are the identity matrix and the all-ones matrix, respectively. Since S 0 (G) = A (G ¯) − A (G) = J − I − 2 A (G) is the Seidel matrix of G , this new matrix is the generalization of Seidel matrix. Further, S α (G) = (1 − α) J + (α n − 1) I − 2 α D (G) − 2 (1 − α) A (G) is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, S α -eigenvalues and S α -energy of G , respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of oriented triangular faces F , the notion of edges potential P E and the notion of triangular faces potential P F in a connected oriented locally finite graph (V , E) in order to construct a new framework that's we call the magnetic weighted 2 -simplicial complex M = (V , E , F , P E , P F). On this new magnetic weighted framework, we introduce the magnetic 0 -cochains set, the magnetic 1 -cochains set, the magnetic 2 -cochains set and the magnetic cochains set. After that, we construct the magnetic 0 -cochains Laplacian, the magnetic 1 -cochains Laplacian, the magnetic 2 -cochains Laplacian and the magnetic cochains Laplacian. Finally, we ensure essential self-adjointness for our new magnetic cochains Laplacians using the Stieltjes vectors. [ABSTRACT FROM AUTHOR]

Let D = (V , A) be a finite simple directed graph (shortly digraph). A function f : V → { − 1 , 1 , 2 , 3 } is called a twin signed double Roman dominating function (TSDRDF) if (i) every vertex v with f (v) = − 1 has at least two in-neighbor assigned a 2 or at least an in-neighbor w with f (w) = 3 , also at least two out-neighbor assigned a 2 or at least an out-neighbor w with f (w) = 3 (ii) every vertex v with f (v) = 1 is adjacent to at least an in-neighbor and an out-neighbor w with f (w) ≥ 2 and (iii) f (N − [ v ]) ≥ 1 and f (N + [ v ]) ≥ 1 hold for any vertex v. The weight of a TSDRDF f is ω (f) = ∑ u ∈ V (D) f (u) , the twin signed double Roman domination number γ sdR ∗ (D) of D is the minimum weight of a TSDRDF on D. In this paper, we initiate the study of twin signed double Roman domination in digraphs and we present some bounds for γ sdR ∗ (D). [ABSTRACT FROM AUTHOR]

In this paper, the notions of ideals in NBCK-algebras are presented and some properties of such ideals are analyzed. Essentially, our approach is to define NBCK-algebras of all cosets by considering an operation on the set of all cosets defined via ideals. Afterwards, our aim is to study ideal theorems on NBCK-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we construct a weighted geometric realization of the set of 2-simplexes and 3-simplexes. On this weighted geometric realization, we create the Laplacian associated to 2-simplexes L S 2 and the Laplacian associated to 3-simplexes L S 3 . We prove that the nonzero spectrum of L S 2 is the same as the nonzero spectrum of L S 3 . For 0, we show that 0 belongs to the spectrum of L S 2 or to the spectrum of L S 3 . [ABSTRACT FROM AUTHOR]

For a d -regular graph G , an edge-signing σ : E (G) → { − 1 , 1 } is called a good signing if the absolute eigenvalues of its adjacency matrix are at most 2 d − 1 . Bilu and Linial conjectured that for each regular graph there exists a good signing. In this paper, by using the concept of "Equitable Partition", we prove the conjecture for some cases. We show how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G , then we can find a good signing for a 2-lift of G. [ABSTRACT FROM AUTHOR]

In this paper, we explore a variant of the game of Cops and Robber introduced by Bonato et al. where the robber is invisible unless outside the common neighborhood of the cops. The hyperopic cop number is analogous to the cop number and we investigate bounds on this quantity. We define a small common neighborhood set and relate the minimum cardinality of this graph parameter to the hyperopic cop number. We consider diameter 2 graphs, particularly the join of two graphs, as well as Cartesian products. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the new concept detour global domination number for splitting graph of standard graph. The detour global dominating sets in some standard and special graphs are determined. First we recollect the concept of splitting graph of a graph and we produce some results based on the detour global domination number of splitting graph of star graph, bistar graph and complete bipartite graph. A set S is called a detour global dominating set of G if S is both detour and global dominating set of G. The detour global domination number is the minimum cardinality of a detour global dominating set in G. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number  γ (G) of G is the minimum cardinality taken over all dominating sets of  G. A dominating set S is called a connected dominating set if the induced subgraph 〈 S 〉 is connected. The minimum cardinality taken over all connected dominating sets of G is called the connected domination number of G and is denoted by γ c (G). In this paper, we give the exact values of domination parameters for edge cycle graphs G (C k) where G = P n , C n , K n and k ∈ { 3 , 4 }. Also, we obtain bounds for the domination parameters of edge cycle graphs G (C k) where G is an arbitrary graph and k ∈ { 3 , 4 } and characterize the extremal graphs. Moreover, Nordhaus–Gaddum-type results are presented for the domination parameters of edge cycle graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a simple graph of order n. A set S ⊆ V (G) is said to be a restrained set if each vertex in V (G) − S is adjacent to a vertex in V (G) − S. The restrained polynomial of a graph G is R (G , x) = ∑ i = 0 n r (G , i) x i where r (G , i) is the number of restrained sets of G of cardinality i. A set S ⊆ V (G) is said to be a restrained dominating set if each vertex in V (G) − S is adjacent to a vertex in S and to a vertex in V (G) − S. The restrained domination polynomial of a graph G is defined by D r (G , x) = ∑ i = 1 n d r (G , i) x i where d r (G , i) is the number of restrained dominating sets of G of cardinality i. In this paper, we derive restrained domination polynomial of join and corona of any two graphs. [ABSTRACT FROM AUTHOR]

A restrained Italian dominating function (RIDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 } satisfying: (i) f (N (u)) ≥ 2 for every vertex u ∈ V (G) with f (u) = 0 , where N (u) is the set of vertices adjacent to u ; (ii) the subgraph induced by the vertices assigned 0 under f has no isolated vertices. The weight of an RIDF is the sum of its function values over the whole set of vertices, and the restrained Italian domination number γ r I (G) is the minimum weight of an RIDF on G. In this paper, we initiate the study of the restrained Italian bondage number b r I (G) of a graph G with no isolated vertices defined as the smallest size of set of edges F ⊆ E (G) for which γ r I (G − F) > γ r I (G). We begin by showing that the decision problem associated with the restrained Italian bondage problem is NP-hard. Then basic properties of the restrained Italian bondage number are presented. Finally, some sharp bounds for b r I (G) are also established. [ABSTRACT FROM AUTHOR]

In this paper, we characterized the k -cost effective dominating sets in the corona and lexicographic product of graphs. Further, we give some formulas for the corona of graphs and upper bound for the lexicographic product of graphs. [ABSTRACT FROM AUTHOR]

Let H = (V (H) , E (H)) be a graph. A k -coloring of H is a mapping π : V (H) → { 1 , 2 , ... , k } , if each color class induces a K 2 -free subgraph. For a graph G of order at least 2 , a G -free k -coloring of H , is a mapping π : V (H) → { 1 , 2 , ... , k } , so that the induced subgraph by each color class of π , contains no copy of G. The G -free chromatic number of H , is the minimum number k , so that it has a G -free k -coloring, and denoted by χ G (H). Suppose that be a family of graphs, we say a graph H has a -free k -coloring, if there exists a map π : V (H) → { 1 , 2 , ... , k } , such that each color class V i = π − 1 (i) does not contain any members of G. In this paper, we give some bounds and attributes on the G -free chromatic number of graphs in terms of the number of vertices, maximum degree, minimum degree, and chromatic number. Our main results are the Nordhaus–Gaddum-type theorem for the -free chromatic number of a graph. [ABSTRACT FROM AUTHOR]

A dominating set D ⊆ V (G) is called a connected co-independent dominating set of G , where G is a finite simple undirected graph, if D is a connected dominating set of G and V (G) ∖ D is an independent set. The minimum cardinality of such a set, denoted by γ c , coi (G) , is called the connected co-independent domination number of G. In this paper, we study this concept and the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the connected co-independent dominating sets in these types of graphs and determine the exact values of their connected co-independent domination numbers. [ABSTRACT FROM AUTHOR]

For an ordered k -decomposition = { G 1 , G 2 , ... , G k } of a connected graph G = (V , E) , the -representation of an edge e is the k -tuple γ (e /) = (d (e , G 1) , d (e , G 2) , ... , d (e , G k)) , where d (e , G i) represents the distance from e to G i . A decomposition is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k -decomposition is its decomposition dimension dec (G). In this paper, decomposition dimension of Cartesian product of paths, cycles and stars is studied. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G 1 and G 2 , let us denote the subdivision-vertex join and subdivision-edge join by G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]

In a graph G = (V , E) , a module is a vertex subset M of V such that every vertex outside M is adjacent to all or none of M. For example, ∅ , { x } (x ∈ V) and V are modules of G , called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex x of a prime graph G is critical if G − x is decomposable. Moreover, a prime graph with k noncritical vertices is called (− k) -critical graph. A prime graph G is k -minimal if there is some k -vertex set X of vertices such that there is no proper induced subgraph of G containing X is prime. From this perspective, Boudabbous proposes to find the (− k) -critical graphs and k -minimal graphs for some integer k even in a particular case of graphs. This research paper attempts to answer Boudabbous's question. First, we describe the (− k) -critical tree. As a corollary, we determine the number of nonisomorphic (− k) -critical tree with n vertices where k ∈ { 1 , 2 , ⌊ n 2 ⌋ }. Second, we provide a complete characterization of the k -minimal tree. As a corollary, we determine the number of nonisomorphic k -minimal tree with n vertices where k ≤ 3. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacency matrix of a graph G. Let s (v) denote the row entries of A (G) corresponding to the vertex v of G. The Hamming distance between the strings s (u) and s (v) is the number of positions in which s (u) and s (v) differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of G ⊕ S. [ABSTRACT FROM AUTHOR]

In this paper, we associate a simple graph to a module over a commutative ring and call it extended graph, that adjacent condition depends on the properties of annihilator submodules. We determine the structure of modules, when the associated graph is star or complete. Moreover, we look for relationships between the extended graph and the zero-divisor graph of modules. [ABSTRACT FROM AUTHOR]

Matrix product (MP) codes over a finite field q , where q a prime power, were first introduced by Blackmore and Norton [MP codes over q , Appl. Algebra Engrg. Comm. Comput. 12 (2001) 477–500]. This paper investigates linear complementary dual (LCD) MP codes over finite commutative Frobenius rings. We derive a necessary and sufficient condition for an MP code to have an LCD. We provide some efficient constructions of LCD MP codes. Finally, we have shown an application on cryptography of LCD codes. [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]

The chromatic number, χ (G) of a graph G is the minimum number of colors used in a proper coloring of G. In an improper coloring, an edge u v is bad if the colors assigned to the end vertices of the edge is the same. Now, if the available colors are less than that of the chromatic number of graph G , then coloring the graph with the available colors leads to bad edges in G. In this paper, we use the concept of δ (k) -coloring and determine the number of bad edges in generalized Petersen graph (P (n , t)). The number of bad edges which result from a δ (k) -coloring of G is denoted by b k (G). [ABSTRACT FROM AUTHOR]

The non-commuting graph Γ G of a finite group G has vertex set as G ∖ Z (G) and any two vertices x , y are adjacent if x y ≠ y x. In this paper, we have determined the metric dimension and resolving polynomial of Γ D n , where D n is the dihedral group of order 2 n. The distance spectrum of Γ D n has also been determined for all n > 2. [ABSTRACT FROM AUTHOR]

Regarding the 2- d i s t a n c e coloring of planar graphs, in 1977, Wegner conjectured that for a graph G : (1) χ 2 (G) ≤ 7 if Δ (G) = 3. (2) χ 2 (G) ≤ Δ (G) + 5 if 4 ≤ Δ (G) ≤ 7. (3) χ 2 (G) ≤ ⌊ 3 2 Δ (G) ⌋ + 1 if Δ (G) ≥ 8. In this paper, we proved that for every planar graph with maximum degree Δ (G) ≤ 4 : (1) χ 2 l (G) ≤ 1 0 if g (G) ≥ 6. (2) χ 2 l (G) ≤ 7 if g (G) ≥ 1 0. [ABSTRACT FROM AUTHOR]

A two-valued function f : V → { − 1 , + 1 } defined on the vertices of a graph G = (V , E) is a majority bad dominating function (MBDF), if the sum of its function values over at least half the closed neighborhoods is at most one. That is, for at least half the vertices v ∈ V , f (N [ v ]) ≤ 1 , where N [ v ] consists of v and every vertex adjacent to v. The majority bad domination number of a graph G, is denoted by β maj (G) , and is defined as β maj (G) = max { w (f) | f   is a  MBDF of  G }. In this paper, we determine some results on majority bad number. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

For a graph G, ω (G) , χ (G) represent the clique number and the chromatic number of G, respectively. A hereditary family of graphs is called χ -bounded with χ -binding function f if χ (G) ≤ f (ω (G)) for all G ∈ . A result of Schiermeyer shows that the class of (P 5 , d a r t) -free graphs has a χ -binding function f (ω) = ω 2 [L. Esperet, L. Lemoine, F. Maffray and G. Morel, The chromatic number of (P 5 , K 4) -free graphs, Discrete Math. 313 (2013) 743–754]. In this paper, we prove that the class of (P 5 , d a r t) -free graphs has a χ -binding function f (ω) = 3 4 ω 2 . For the class of (P 5 , C 5 , d a r t) -free graphs, we give a χ -binding function f (ω) = ω + 1 2 . [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , an edge connectivity λ is the minimum number of edges whose removal disconnects G. In this paper, we define property restricted edge connectivity. Then we appraise its particular case, more precisely k-restricted edge connectivity. Some of the basic properties of k-restricted edge connectivity are considered. We also discuss how k-restricted edge connectivity associates with graph coloring. In particular, we try to study how k-restricted edge connectivity relates with Tait's coloring theorem and thereby the famous Four-color conjecture. [ABSTRACT FROM AUTHOR]

The k-2-distance coloring of graph G is a mapping c : V (G) → { 1 , 2 , ... , k } if any two vertices at distance at most two from each other get different colors. The 2-distance chromatic number is the smallest integer k such that G has a k-2-distance coloring, denoted by χ 2 (G). In this paper, we prove that every planar graph G without 3-cycles and intersecting 4-cycles and Δ (G) ≥ 8 1 has χ 2 (G) ≤ Δ + 5. [ABSTRACT FROM AUTHOR]

Among the unsolved problems of enumeration of graphs, one of the most difficult is the problem of enumeration of transitive digraphs, which is equivalent to the problem of enumeration of finite partial orders and is equivalent to the problem of enumeration of finite labeled T 0 -topologies. The sequence { T 0 (n) } , where T 0 (n) is the number of all labeled T 0 -topologies defined on an n-set, has been studied from different points of view. In the previous work of the second author, a formula was obtained in which the number T 0 (n) is represented as a linear combination of numbers W (p ¯) (where the sequences p ¯ = (p 1 , ... , p k) ⊧ n are compositions of the number n). Recurrent relations between individual numbers W (⋅) were obtained. In this paper, new recursive formulas between separate numbers W (⋅) are obtained. [ABSTRACT FROM AUTHOR]

In secure domination [A. P. Burger, M. A. Henning and J. H. Van Vuuren, Vertex covers and secure domination in graphs, Quaest Math. 31 (2008) 163–171; E. J. Cockayne, Irredundance, secure domination and maximum degree in trees, Discrete Math. 307(1) (2007) 12–17; E. J. Cockayne, O. Favaron and C. M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003) 87–100; C. M. Mynhardt, H. C. Swart and E. Ungerer, Excellent trees and secure domination, Util. Math. 67 (2005) 255–267], a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with equitability. Secure equitable dominating set is introduced and studied. Other concepts like independence and rigid security are also studied in this paper. [ABSTRACT FROM AUTHOR]

An edge labeling of a connected graph G = (V , E) is said to be local antimagic if it is a bijection f : E → { 1 , ... , | E | } such that for any pair of adjacent vertices x and y, f + (x) ≠ f + (y) , where the induced vertex label f + (x) = ∑ f (e) , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G) , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with χ l a = 2 , 3. [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and G be a simple graph with vertex set V (G). Let f be a function that assigns label from the set { 0 , 1 , 2 , ... , k + 1 } to the vertices of a graph G. For a vertex v ∈ V (G) , the active neighborhood of v , denoted by AN (v) , is the set of vertices w ∈ N G (v) such that f (w) ≥ 1. A [ k ] -RDF is a function f : V (G) → { 0 , 1 , 2 , ... , k + 1 } satisfying the condition that for any vertex v ∈ V (G) with f (v) < k , f (N G [ v ]) ≥ | AN (v) | + k. The weight of a [ k ] -RDF is ω (f) = Σ v ∈ V (G) f (v). The [ k ] -Roman domination number γ [ k R ] (G) of G is the minimum weight of an [ k ] -RDF on G. The case k = 4 is called quadruple Roman domination number. In this paper, we first establish an upper bound for quadruple Roman domination number of graphs with minimum degree two, and then we derive a Nordhaus–Gaddum bound on the quadruple Roman domination number of graphs. [ABSTRACT FROM AUTHOR]

A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this paper, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitly the total dominator chromatic number of the middle graph of several known families of graphs. [ABSTRACT FROM AUTHOR]

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I m-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II m-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n, the normalized Laplacian is a square matrix of order n, defined as ℒ (G) = D (G) − 1 2 L (G) D (G) − 1 2 , where D (G) − 1 2 is the diagonal matrix whose i-th diagonal entry is 1 d i . In this paper, we find the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of quotient matrix associated with graph G. For a finite group , the power graph () of a group is defined as the simple graph in which two distinct vertices are joined by an edge if and only if one is the power of the other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of the power graphs of the finite cyclic group ℤ n . [ABSTRACT FROM AUTHOR]

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round, the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we examine whether the Mycielskian of G, μ (G) , is k-indicated colorable for all k ≥ χ i (μ (G)) , whenever G is l-indicated colorable for all l ≥ χ i (G). In this direction, we prove that the Mycielskian of the bipartite graphs, complete multipartite graphs, { P 5 , K 3 } -free graphs, { P 5 , K 4 , K i t e , B u l l } -free graphs, { 2 K 2 , C 4 } -free graphs and { P 6 , C 5 , P 5 ¯ , K 1 , 3 } -free graphs are k-indicated colorable for all k greater than or equal to the indicated chromatic number of the corresponding graphs. [ABSTRACT FROM AUTHOR]

A graph G = G(V,E) is called L-list colorable if there is a vertex coloring of G in which the color assigned to a vertex v is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(v) have the cardinality k and G is L-list colorable for all possible assignments of such lists. A graph G is said to be chromatic choosable if its chromatic and list chromatic numbers are equal. Investigation of χ-choosable graphs is one of the open problems. In this paper, we investigate this problem for classes of perfect graphs. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a graph G is said to be locally identifying (lid-coloring) if for any pair u,v of adjacent vertices with distinct closed neighborhoods, the sets of colors in the closed neighborhoods of u and v are different. The smallest integer k for which G admits a lid-coloring is called the lid-chromatic number of G. The corona product G ⊙ H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H for every i = 1, 2,…,|V (G)|, where |A| denotes the number of elements in the set A. In this paper, the lid-chromatic number of corona product of graphs has been studied. [ABSTRACT FROM AUTHOR]

In this paper, we illustrate the rule for finding number of idempotents in the doubly stochastic matrix semigroup Dn and also locate the idempotents in the case of D3 and D4. Further, we describe idempotent generated ideals of these semigroups and it is shown that the idempotent generated ideals form a lattice. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥⋯ ≥ μn−1 > μn = 0, the Laplacian resolvent energy of G is defined as RL(G) =∑i=1n 1 n+1−μi. In this paper, we provide an improved lower bound for RL(G) through majorization. Considering our lower bound, we also derive some lower bounds for RL(G) when the graph G possesses tree structure. [ABSTRACT FROM AUTHOR]

Given a set ℋ of some positive integers, a positive integer n is ℋ-testable if there are two disjoint subsets P and Q of ℋ such that n =∑p∈Pp −∑q∈Qq. The testability of the set Nn of positive integers not exceeding n, denoted by τ(n), is the minimum integer m such that there is a set ℋ with m positive integers satisfying that each t ∈ Nn is ℋ-testable. In this short paper, we prove that τ(n) = ⌈log3(2n + 1)⌉ and propose an algorithm to output the representation of an arbitrary positive integer by the form ∑i∈X3i −∑ j∈Y3j, where X and Y are two disjoint sets of some non-negative integers not exceeding τ(n). [ABSTRACT FROM AUTHOR]

Let Γ be a simple graph of order n having m edges and let Γ← be an orientation of Γ. The hermitian skew Laplacian matrix SL(Γ←) of Γ← is defined as SL(Γ←) = D¯(Γ←) − iS(Γ←), where i = −1 is the imaginary unit, D¯(Γ←) is the diagonal matrix of oriented degrees d¯i = di+ − d i− and S(Γ←) is the skew matrix. The spectral radius of the matrix SL(Γ←) is the hermitian skew Laplacian spectral radius and the sum of the absolute values of eigenvalues of SL(Γ←) is the hermitian skew Laplacian energy of Γ←. In this paper we study the hermitian spectral radius and hermitian skew Laplacian energy of Γ←. We obtain some new bounds for the hermitian skew Laplacian spectral radius and characterize the extremal oriented graphs attaining these bounds. Further, we obtain some bounds for the hermitian skew Laplacian energy and locate the extremal oriented graphs. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm is present that uses the sum of disjoint products (SDP) to calculate the reliability bounds of a complex system. The algorithm requires a set of all minimal paths and cuts to compute the bounds of the reliability polynomial. Furthermore, using Boolean algebra to construct a sum-of-products expression. This research aims to create the lower and upper bounds of reliability for a complex communication system (Double-bridge). By minimizing the number of disjointed (mutually exclusive) variables, the method can reduce the computer time required to estimate the probability of the principal events. The communication system is also used to establish the usefulness of the provided algorithm, which is better than the currently used SDP method for big data sets and more specific to understand and implement. Furthermore, simulations and numerical examples have shown that the new technique typically improves Spross bounds, particularly for highly reliable systems. [ABSTRACT FROM AUTHOR]

The deficiency df(v) of a vertex v is the deviation between the degree of the vertex v and the maximum degree of the graph. The maximum deficiency matrix Mdf(G) of a simple graph G is square matrix whose (i,j)th entry is max(df(vi),df(vj)), if the vertices vi and vj are adjacent and 0, otherwise. Maximum deficiency energy EMdf(G) is the absolute sum of eigenvalues of maximum deficiency matrix Mdf(G). In this paper, we find maximum deficiency energy of some classes of graphs. Moreover, we construct an algorithm and Python 3.8 program to find out spectrum and maximum deficiency energy of G. [ABSTRACT FROM AUTHOR]

Generalized hypercube network is an interconnection network topology. The topology of interconnection network usually takes graph as a mathematical model, in which the vertex of graph represents the server and the edge of graph represents the connection between servers. As vertex count increases, the vertex degree of hypercube network increases as well. This has advantages, such as increased performance and reliability. The domination parameters are an important basis for analyzing and measuring the reliability of interconnection networks. In this paper, few properties of generalized hypercube network and Roman detour domination number for minimum dimensional generalized hypercube networks and hypercube networks are discussed. And also we determine the bounds for γRd(Q(d1,d2,…,dn)) and γRd(Qn). [ABSTRACT FROM AUTHOR]

A bijective product square k-cordial labeling of a graph G with vertex set V and edge set E is a bijection f from V (G) to {1, 2,…,|V (G)|} such that the induced edge function f∗ from E(G) to Sk = {i2(modk), 0 ≤ i ≤ k − 1} defined as f∗(uv) ≡ (f(u)f(v))2(modk) for every edge uv ∈ E(G) satisfies the following condition. (i) If ef∗(i) is the number of edges labeled with i under f∗, then |ef∗(i) − e f∗(j)|≤ 1 where i,j ∈ Sk. A graph which admits a bijective product square k-cordial labeling is called bijective product square k-cordial graph. In this paper, we find upper limit of the size of the bijective product square k-cordial graph for k = 3 and 5. Also, we prove that every graph is a subgraph of a connected bijective product square k-cordial graph for all odd prime k. In addition, we establish the relation between bijective product square k-cordial and k-product cordial graphs. [ABSTRACT FROM AUTHOR]