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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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Several reasonably cyclotomic sequences are constructed by cyclotomic classes having good pseudo-randomness property. In this paper, we derive the linear complexity of an extended binary cyclotomic sequences of order 2 l t over finite field having period p q. Our result shows that these sequences have higher linear complexity, which can resist linear attack. [ABSTRACT FROM AUTHOR]

A bar visibility representation of a graph G is an assignment of the vertices of G to distinct horizontal line segments in the plane so that two vertices are adjacent in G if and only if there is an uninterrupted vertical channel of positive width that joins the bars corresponding to those vertices. A bar visibility representation is called a unit bar visibility representation if every bar has the same length. If each vertex is assigned to t distinct bars of the same length in a unit bar visibility representation then the representation is called t -unit bar visibility representation. In this paper, we introduce a "sliding column model" for t -unit bar visibility representation and show that every graph of maximum degree Δ has a t -unit bar visibility representation for t ≤ ⌊ Δ + 1 2 ⌋. We also show that a planar graph of maximum degree 3 having n vertices and m edges has a 2-unit bar visibility representation on 2 n − m columns and a 3-connected cubic graph of n vertices admits a 2-unit bar visibility representation on n 2 columns. [ABSTRACT FROM AUTHOR]

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

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

Let G be a finite undirected simple connected graph with vertex set V (G) and edge set E (G). A vertex u ∈ V (G) resolves two elements (vertices or edges) v , w ∈ V (G) ∪ E (G) if d (u , v) ≠ d (u , w). A subset S m of vertices in G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertices of S m . The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim m (G). In this paper, we study the mixed metric dimension for the plane graph of web graph n and convex polytope n . [ABSTRACT FROM AUTHOR]

The network components of interconnection networks are exposed to failures. But it is necessary that a network should not break down when such failures occur. Thus reliability and efficiency of interconnection networks have been the subject of extensive study in past years. The connectivity, wide diameter, Rabin number, and fault diameter of a network are used as measures of reliability and efficiency. In this paper, the 3 -wide diameter, 3 -Rabin number, and 1 -fault diameter of brother tree BT (s) , s ≥ 2 are obtained. [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]

Metric-based parameters aided the study of the structural properties of graph network space like complexity, central tendency, connectivity, robustness, accessibility, and vulnerability. At the same time, these parameters played a pivotal role in rectifying problems in navigation, integer programming, optimization, pattern recognition and avoidance, combinatoric detection, backtracking in geographical routing, etc. In this paper, we considered the fractional edge dimension, a metric-based parameter in local environment by initiating the study of the fractional local edge dimensions of a graph (FLED). We define the aforesaid concept and give its bounds. We discussed the characterization problem for FLED equal to 1 , the realization problem, and its relation with the fractional metric dimensions. We also calculated the FLED of the Petersen graph, Coxeter graph, the families of cycle, complete, wheel, grid and complete r -partite graphs. [ABSTRACT FROM AUTHOR]

An edge labeling of a 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 of x is f + (x) = ∑ e ∈ E (x) f (e) (E (x) is the set of 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, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs. [ABSTRACT FROM AUTHOR]

Chen et al. [On coupon coloring of graphs, Discrete Appl. Math. 193 (2015) 94–101] had introduced the concept of coupon coloring for any graph with no isolated vertex. A k -coupon coloring of G is an assignment of colors from [ k ] : = { 1 , 2 , ... , k } to the vertices of G such that the neighborhood of every vertex of G contains vertices of all colors from [ k ]. The maximum k for which a k -coupon coloring exists is called the coupon coloring number of G , and is denoted by χ c (G). In this paper, we determine the coupon coloring number of Kneser graphs K (n , 2). [ABSTRACT FROM AUTHOR]

In this paper, we study mean–variance–Conditional Value-at-Risk (CVaR) portfolio optimization problem with short selling, cardinality constraint and transaction costs. To tackle its mixed-integer quadratic optimization model for large number of scenarios, we take advantage of the penalty decomposition method (PDM). It needs solving a quadratic optimization problem and a mixed-integer linear program at each iteration, where the later one has explicit optimal solution. The convergence of PDM to a partial minimum of original problem is proved. Finally, numerical experiments using the S&P index for 2020 are conducted to evaluate efficiency of the proposed algorithm in terms of return, variance and CVaR gaps and CPU times. [ABSTRACT FROM AUTHOR]

Two mathematical fields that concentrate on creating and analyzing structures are algebra and graph theory. There are numerous studies linking algebraic structures like groups, rings, fields and vector spaces with graph theory. Several algebraic graphs have been defined based on the properties of the order of the group and its elements. In this paper, we systematically review the literature on such graphs to understand the research dynamics in the field. [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]