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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]

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]

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]

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]

A decomposition of a graph is a collection of its edge — disjoint subgraphs whose union equals the edge set of the graph. A decomposition all of whose members are paths is called a path decomposition. In this paper, we introduce and initiate a study of a new variation of decomposition namely equitable path decomposition of a graph G which is defined to be a decomposition in which all the members are paths and the lengths of any two paths are differ by at most one in G. In this paper, we initiate a study on this parameter. [ABSTRACT FROM AUTHOR]

Let ℤn be the ring of integer modulo n with two binary operators, addition (+) and multiplication (.), where n is a positive integer. The special set 풮 is defined as 풮 = {a ∈ ℤn : (∃b ∈ ℤn) ba ≡ a, b≢a, b≢1}. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set 풮 and denote it by Ω−(Z n). In this paper, we define a relationship between G and Ge∗, Ge∗ is a derived graph from G by removing r edges, where r is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and Ω−(Z n). We also provide values of n for which the graph Ω−(ℤ n) is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness. [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]

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]

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 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]

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]

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]

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]

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]

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over ℤ2b, ℤ2pandℤp which have applications in error detection and error correction. [ABSTRACT FROM AUTHOR]

Cayley graph is a graph that encodes the abstract structure of a group. It gives a way of encoding information about a group in a graph. On the other hand, hypergroup is a generalization of group in which the composition of any two elements is a non-empty set. The purpose of this paper is to find a suitable generalization of Cayley graphs to cover hypergroups. More precisely, we introduce generalized Cayley graphs over hypergroups, study their properties and find a simple tool to construct large connected GCH-graphs from smaller GCH-graphs by using graph product. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 and Un be the multiplicative group of the additive group ℤn of integers modulo n. The unitary Haar graph Hn = Haar(ℤn,Un) is a graph with vertex set ℤn ×{0, 1} and the edge set {{(a, 0), (b, 1)}|b − a ∈ Un}. In this paper, we study some structural properties of Hn such as total chromatic number, diameter, planarity, the girth, vertex-connectivity and edge-connectivity. Also, some spectral properties of the graph Hn are determined. [ABSTRACT FROM AUTHOR]

Luo, Tian and Wu [Discrete Math. 345(4) (2022) 112788] conjectured that for any tree T with bipartition (X,Y ), every k-connected bipartite graph G with minimum degree at least k + w, where w =max{|X|,|Y |}, contains a tree T′≅T such that κ(G − V (T′)) ≥ k. In the paper, we confirm the conjecture when T is an odd path on m vertices. We remind that Yang and Tian [Proof of a conjecture on connectivity keeping odd paths in k-connected bipartite graphs, arXiv:2209.08373v2] also prove the same result by a different way. [ABSTRACT FROM AUTHOR]

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points (n) asymptotically approaches a constant (βTSP) as n becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size (n) comprise a normal distribution whose mean approaches βTSP as n increases. Moreover, as n approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of βTSP is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and βTSP, based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data. [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 χi(G).In this paper, we observe that the Nordhaus–Gaddum inequalities for the chromatic number are also satisfied by the indicated chromatic number. Further, we prove that outerplanar graphs, 3-colorable maximal planar graphs with col(G)≠5, uniquely 4-colorable planar graphs and Mn (obtained from Cn by joining diagonally opposite vertices) are k-indicated colorable for all k greater than or equal to its chromatic number. Also, we show that Ben wins the game with 4 colors when Ann uses any connected strategy on M4t+1 for any t ≥ 2. This turns out to be a generalization of the result due to Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472]. Finally, we prove that Mycielskian of G is (k + 1)-indicated colorable when G is k-indicated colorable for some k ≥ χi(G). These partially answer one of the questions which was raised by Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472]. [ABSTRACT FROM AUTHOR]

As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In [A. Miličević, A. Nikolić and S. Trinajstić, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393–399], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and found the formulas of the F-index, hyper-Zagreb index, and Sigma index for double join of graphs based on the total graph and double corona of graphs based on the total graph. Moreover, some applications of the derived results are also discussed. [ABSTRACT FROM AUTHOR]

Let 픽ql be a finite field of cardinality ql, where q is a power of a prime and l is a prime integer. An 픽q-linear code over 픽ql of length n is an 픽q-subspace of 픽qln. If an 픽q-linear code over 픽ql with parameters (n,qk,d H) meets the Singleton bound, i.e., dH = n −⌈k l ⌉ + 1, then it is called an MDS code. In this paper, by use of a Fourier matrix and a canonical form decomposition of constacyclic 픽q-linear codes over 픽ql, we construct some optimal 픽q-linear codes over 픽ql which have the same parameters as the MDS codes over 픽ql. [ABSTRACT FROM AUTHOR]

Inspired by the studies on conditional connectivity by Harary [1], we worked on k-restricted edge connectivity of a graph G [4]. The k-restricted edge connectivity of a graph G is the cardinality of a minimum edge-cut whose deletion disconnects the graph G into components where each component has order at least k. In this work, we take into account some results on k-restricted edge connectivity and the novel parameter exact k-restricted edge connectivity. In the definition of k-restricted edge connectivity, we impose some extra constraints to achieve this parameter. In this paper, we consider some fundamental aspects of exact k-restricted edge connectivity and how they relate to the k-restricted edge connectivity. [ABSTRACT FROM AUTHOR]

The nonzero component graph of a finite-dimensional vector space over a finite field is a graph whose vertices are the nonzero vectors in the vector space, and any two vertices are adjacent if the corresponding linear combination contains a common basis vector. In this paper, we discuss the centrality measures and entropy of the nonzero component graph and also analyze the sensitivity of the graph using the centrality measures. [ABSTRACT FROM AUTHOR]

An incidence of a graph G is a pair (u,e) where u is a vertex of G and e is an edge of G incident with u. Two incidences (u,e) and (v,f) of G are adjacent whenever (i) u = v, or (ii) e = f or (iii) uv = e or f. A strong incidence coloring of a graph G is a mapping from the set of incidences of G to the set of colors {1,…,k}, such that every two incidences that are adjacent, or adjacent to the same incidence receive distinct colors. In this paper, we prove that every connected subcubic graph G except K3,3 has a strong incidence coloring with at most 17 colors. [ABSTRACT FROM AUTHOR]

For a graph G, let ik(G) be the number of independent sets of size k in G. The independence polynomial I(G; x) =∑k≥0ik(G)xk has been the focus of considerable research. In this paper, using the coefficients of independence polynomials, we order graphs with some given parameters. We first determine the extremal graph whose all coefficients of I(G; x) are the largest (respectively, smallest) among all connected graphs (respectively, bipartite graphs) with given vertex cover number. Then we also derive the extremal graph whose all coefficients of I(G; x) are the largest among all connected graphs (respectively, bipartite graphs) with given vertex dominate number. [ABSTRACT FROM AUTHOR]

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 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 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]

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]

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]

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]

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]

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]

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]

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]

We consider a new coloring as follows such that a subset of V (G) needs very few colors. Let G be a graph with vertex set V (G) and a,b,i be three integers with a ≥ b ≥ i ≥ 0. S is a vertex subset of G, graph G is said to be i-weak (a : b)-choosable about set S, if the following can be satisfied: For any list assignment L = {L(v) : |L(v)| = a,v ∈ V (G)}, there is a way that each vertex v of S is assigned i colors of L(v) and each vertex u of V (G) ∖ S is assigned b colors of L(v) such that the color sets corresponding to adjacent vertices do not intersect. In this paper, we consider the i-weak (a : b)-choosable problem of 1-planar graphs without short cycles and prove our results by the discharging method. In this paper, we prove that every 1-planar graph without 3-cycles, 5-cycles and adjacent 4-cycles is i-weak (4b + i : b)-choosable about set S = {v : v ∈ V (G),d(v) = 4}. [ABSTRACT FROM AUTHOR]

The graphs used in this paper are simple, connected and finite graph. A graceful m-coloring of a graph G is a proper vertex coloring c : V (G) →{1, 2,m}, where m ≥ 2 which induces a proper edge coloring c′ : E(G) →{1, 2,m − 1} that defined by c′(uv) = |c(u) − c(v)|. Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful m-coloring for m ∈ N. The minimum vertex coloring, that can apply such that a graph G can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by χg(G). In this paper, we will investigate the graceful chromatic number of generalized Petersen graph P(n,k) for k = 1, 2, 3, 4. [ABSTRACT FROM AUTHOR]

The rings considered in this paper are commutative with identity which are not integral domains. Let R be a ring. An ideal I of R is said to be an annihilating ideal if there exists r ∈ R∖{0} such that Ir = (0). Let 픸(R) denote the set of all annihilating ideals of R and we denote 픸(R)∖{(0)} by 픸(R)∗. With R, in this paper, we associate an undirected graph denoted by SΩ(R) whose vertex set is 픸(R)∗ and two distinct vertices I,J are adjacent in this graph if and only if either IJ = (0) or I + J ∈ 픸(R). The aim of this paper is to study the interplay between some graph properties of SΩ(R) and the algebraic properties of R and to compare some graph properties of SΩ(R) with the corresponding graph properties of the annihilating ideal graph of R and the sum annihilating ideal graph of R. [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]

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]

Graph protection using mobile guards has received a lot of attention in the literature. It has been considered in different forms, including Eternal Dominating set, Eternal Independent set and Eternal Vertex Cover set. In this paper, we introduce and study two new models of graph protection, namely Eternal Feedback Vertex Sets (EFVS) and m-Eternal Feedback Vertex Sets (m-EFVS). Both models are based on an initial selection of a feedback vertex set (FVS), where a vertex in FVS can be replaced with a neighboring vertex such that the resulting set is a FVS too. We prove bounds for both the eternal and m-eternal feedback vertex numbers on, mainly, distance graphs, circulant graphs and grids. Also, we deduce other inequalities for both parameters on cycles, complete graphs and complete bipartite graphs. [ABSTRACT FROM AUTHOR]