Vertex Set Of Graph Research Articles

Generalization of Randic ́ Index of the Non-commuting Graph for Some Finite Groups

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.

Topologies induced by graph metrics on the vertex set of graphs

Topologies induced by graph metrics on the vertex set of graphs

Computing topological indices of certain windmill graphs

Let V (G), be the vertex set of graph G, we define Vd = {v ∈ V (G) : SG(u) = d} in which SG(u) = ∑v∈NG(u) dG(v) and NG(u) = {v ∈ V(G) : uv ∈ E(G)}. In this paper we express the explicit formula for Sanskruti index S(G), fth M1 and M2 Zagreb indices, fifth M1 and M2 multiplicative Zagreb indices for Dutch windmill graph, Kulli cyclic windmill graph, Kulli path windmill graph and French windmill graph.

Reliability of the round matching composition networks based on g-extra conditional fault

Connectivity and diagnosability are very important parameters for the tolerant of an interconnection network. Let F be a vertex set of graph G, the g-extra connectivity k˜(g)(G) of G requires each component of G−F has at least (g+1) vertices. The g-extra conditional diagnosability t˜(g)(G) is the maximum number of faulty vertices that the graph can be guaranteed to identify under the condition that each fault-free component has at least g+1 vertices. In this paper, we study the problem about the g-extra connectivity k˜(g)(G) of the round matching composition networks, and obtain some results about the g-extra conditional diagnosability of the round matching composition networks under the PMC model and MM* model, respectively.

The Bounds of Generalized 4-Connectivity of Alternating Group Graphs

Generalized connectivity is a parameter of evaluating the reliability of a network. Let [Formula: see text] be a vertex set of graph [Formula: see text] and [Formula: see text], there is a tree in [Formula: see text] as it connects all vertices of [Formula: see text] which is called a [Formula: see text]-tree. Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-trees in [Formula: see text]. If [Formula: see text] and [Formula: see text] are given, we define that two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint. Then the maximum number of internally disjoint [Formula: see text]-trees in [Formula: see text] is denoted by [Formula: see text]. For an integer [Formula: see text] with [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is denoted as [Formula: see text]. In this paper, we study the generalized [Formula: see text]-connectivity of alternating group graphs, denoted by [Formula: see text]. We show the upper bound and the lower bound of [Formula: see text], [Formula: see text] for [Formula: see text], and that [Formula: see text] is given.

The reflexive edge strength on some almost regular graphs

A function f with domain and range are respectively the edge set of graph G and natural number up to ke, and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to 2kv are called a total k-labeling where k=max{ke,2kv}. The total k-labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k-labeling, where for any edge x1x2, the weight is wt(x1x2)=fv(x1)+fe(x1x2)+fv(x2). The reflexive edge strength of the graph G, denoted by res(G) is the minimum k for graph G which has an edge irregular reflexive k-labelling. In this study, we obtained the res(G) of graphs which their vertex degrees show an almost regularity properties.

On resolving efficient domination number of comb product of special graphs

Let G be a connected, finite, and undirected graph. A vertex set D in G is an efficient dominating set of G if D is an independent set and for each point υ ∈ V(G)-D is adjacent to precisely one vertex d ∈ D. The representation of points υ ∈ V(G) in respect of an ordered set W = {w 1, w 2,…, wk } is the k–vector r(υ| W) = (d(υ, w 1), d(υ, w 2),…, d(v, wk )), which d(u, v) is the distance between the points u and υ. The set W is a resolving set of G if r(u|W) = r(υ| W), for each point u and υ in G. A set of vertices in graph G which is an efficient dominating set and resolving set is called a resolving efficient dominating set. The minimal cardinality of resolving efficient dominating set is called resolving efficient domination number, denoted by γre (G). The comb product between graph G and graph H is a graph which get from taking a copy of graph G as many vertices of graph H and grafting the i-th copy of graph G to each vertex of H, and its notated by G ▹ H. In this paper, we determine the resolving efficient domination number of comb product graph, namely Kn ▹C 3, Kn ▹P 3, Wn ▹C 3, Wn ▹P 3, and Sn ▹P 2

Odd harmonious labeling on squid graph and double squid graph

An injective function f from set of vertices in graph G to a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the function f induced the edge function f* from the set of edges of G to a set of odd positive integer number {1,3,5,…,2|E| − 1} with f*(xy) = f(x) + f(y) for every edge xy in E. Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graph Tn,k is a graph which is obtained from a cycle Cn and we add k pendant to one vertex of the cycle. It is known that Cn is an odd harmonious graph if and only if n = 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graph Tn,k and T 2n,k are an odd harmonious graph, for n = 0 (mod 2), n ≥ 4 and k ≥ 1. The construction of the odd harmonious labeling of the graph Tn,k and T 2n,k are inspired by the odd harmonious labeling of Cn for n = 0(mod 4).

The Topological Indices of the Non-commuting Graph for Symmetric Groups

Topological indices are the numerical values that can be calculated from a graph and it is calculated based on the molecular graph of a chemical compound. It is often used in chemistry to analyse the physical properties of the molecule which can be represented as a graph with a set of vertices and edges. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if they do not commute. The symmetric group, denoted as S_n, is a set of all permutation under composition. In this paper, two of the topological indices, namely the Wiener index and the Zagreb index of the non-commuting graph for symmetric groups of order 6 and 24 are determined. Keywords: Wiener index; Zagreb index; non-commuting graph; symmetric groups

Topological Indices of the Non-commuting Graph for Generalised Quaternion Group

A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties and chemical reactivity. It is calculated from a graph representing a molecule. Meanwhile, the non-commuting graph, $$\Gamma _G$$ of G, is defined as a graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. The main objective of this article is to determine the general formula of some topological indices, namely Wiener index, first Zagreb index and second Zagreb index for the non-commuting graph associated with generalised quaternion group in terms of n.

Some algebraic hyperstructures related to zero forcing sets and forcing digraphs

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph [Formula: see text]. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph [Formula: see text]. Finally, we give some examples to clarify these hyperstructures.

Topological indices of non-commuting graph of dihedral groups

Assume is a non-abelian group A dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. The non-commuting graph of denoted by is the graph of vertex set whose vertices are non-central elements, in which is the center of and two distinct vertices and are joined by an edge if and only if In this paper, some topological indices of the non-commuting graph, of the dihedral groups, are presented. In order to determine the Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graph, of the dihedral groups, previous results of some of the topological indices of non-commuting graph of finite group are used. Then, the non-commuting graphs of dihedral groups of different orders are found. Finally, the generalisation of Edge-Wiener index, First Zagreb index and Second Zagreb index of the non-commuting graphs of dihedral groups are determined.

Constructing Non-Commuting Graph of Symmetric Groups by Using Maple's Software

Maple is a symbolic and numeric computing environment and also a multi-paradigm programming language. It is also covers other aspects of technical computing such as visualisation and matrix computation. Maple can be used to construct the graph related to groups which involves the package of group theory and graph theory. The aim of this paper is to present the Maple’s coding in constructing the non- commuting graph of the symmetric groups. Meanwhile, the non- commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only they do not commute. The symmetric group, denoted by is a group of permutations on a set with elements. This paper only focuses on finding the non-commuting graph of symmetric groups of order six and 24.

Spectral properties of cographs and P5-free graphs

ABSTRACTA cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph G is a cograph if and only if no induced subgraph of G has an eigenvalue in the interval . It is also shown that the multiplicity of any eigenvalue of a cograph G does not exceed the sum of multiplicities of 0 and as eigenvalues of G. We introduce a partial order on the vertex set of graphs G in terms of inclusions among the open and closed neighbourhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph G except for does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we prove that bipartite -free graphs have no eigenvalue in the intervals and (0, 1 / 2).

Local Edge Antimagic Coloring of Comb Product of Graphs

All graph in this paper are finite, simple and connected graph. Let G(V, E) be a graph of vertex set V and edge set E. A bijection is called a local edge antimagic labeling if for any two adjacent edges e1 and e2, , where for . Thus, any local edge antimagic labeling induces a proper edge coloring of G if each edge e is assigned the color w(e). The local edge antimagic hromatic number γlea(G) is the minimum number of colors taken over all colorings induced by local edge antimagic labelings of G. In this paper, we have found the lower bound of the local edge antimagic coloring of and determine exact value local edge antimagic coloring of .

Super local edge antimagic total coloring of

In this paper, we consider that all graphs are finite, simple and connected. Let G(V, E) be a graph of vertex set V and edge set E. A bijection is called a local edge antimagic labeling if for any two adjacent edges e1 and e2, , where for . Thus, any local edge antimagic labeling induces a proper edge coloring of G if each edge e is assigned the color w(e). It is considered to be a super local edge antimagic total coloring, if the smallest labels appear in the vertices. The super local edge antimagic chromatic number, denoted by γleat(G), is the minimum number of colors taken over all colorings induced by super local edge antimagic total labelings of G. In this paper we initiate to study the existence of super local edge antimagic total coloring of comb product of graphs. We also analyse the lower bound of its local edge antimagic chromatic number. It is proved that . Furthermore we have determine exact value local edge antimagic coloring of and .

On the local vertex antimagic total coloring of some families tree

Let G(V, E) be a graph of vertex set V and edge set E. Local vertex antimagic total coloring developed from local edge and local vertex antimagic coloring of graph. Local vertex antimagic total coloring is defined if for any two adjacent vertices v1 and v2, , where for , where E(v) and V(v) are respectively the set of edges incident to v and the set of vertices adjacent to v. Thus, any local vertex antimagic total coloring induces a proper vertex coloring of G if each vertex v is assigned the color w(v). The chromatic number of local vertex antimagic total coloring denote χlvat(G) is the minimum number of colors taken over all colorings induced by local vertex antimagic total coloring of graph G. In this paper, we use some families tree graph. We also study the existence of local vertex antimagic total coloring chromatic number of some families tree namely star graph, double star graph, banana tree graph, centipede graph, and amalgamation of star graph.

Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

Graph Theory Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.

Hereditary Efficiently Dominatable Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C4)‐free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw‐free graphs.

Adaptive Identification of Sets of Vertices in Graphs

Combinatorics In this paper, we consider a concept of adaptive identification of vertices and sets of vertices in different graphs, which was recently introduced by Ben-Haim, Gravier, Lobstein and Moncel (2008). The motivation for adaptive identification comes from applications such as sensor networks and fault detection in multiprocessor systems. We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes.