Closeness and Residual Closeness of Harary Graphs

Analysis of a network in terms of vulnerability is one of the most significant problems. Graph theory serves as a valuable tool for solving complex network problems, and there exist numerous graph-theoretic parameters to analyze the system's stability. Among these parameters, the closeness parameter stands out as one of the most commonly used vulnerability metric. Its definition has evolved over time to enhance ease of formulation and applicability to disconnected structures. Furthermore, based on the closeness parameter, residual closeness, which is a newer and more sensitive parameter compared to other existing parameters, has been introduced as a new graph vulnerability index by Dangalchev. In this study, the outcomes of the closeness and residual closeness parameters in Harary Graphs have been examined. Harary Graphs are well-known constructs that are distinguished by having $n$ vertices that are $k$-connected with the least possible number of edges.Comment: 21 pages preprin

The common-neighbourhood of a graph

The vulnerability measures on a connected graph which are mostly used and known are based on the Neighbourhood concept. Neighbour-integrity, edge-integrity and accessibility number are some of these measures. In this work we define and examine the Common-neighbourhood of a connected graph as a new global connectivity measure. It takes account the neighbourhoods of all   pairs of vertices. We show that, for connected graphs G1 and G2 of order n, if the dominating number of G1 is bigger than the dominating number of G2, then the common- neighbourhood of G1 is less than the common-neighbourhood of G2. We prove some theorems on common-neighbourhood of a graph

INDEPENDENT TRANSVERSAL DOMINATION NUMBER IN COMPLEMENTARY PRISMS

A set D subset of V (G) is an independent transversal dominating set of G if D is a dominating set and also intersects every maximum independent set in G. The minimum cardinality of such a set is equal to the transversal domination number, denoted by gamma(it)(G). This paper is devoted to the computation of the independent transversal domination number of some complementary prism.Ege University MSc Scientific Research Project (BAP) [FYL-2019-20895]This paper is supported by Ege University MSc Scientific Research Project (BAP) under project number FYL-2019-20895

On the bondage, strong and weak bondage numbers in Complementary Prism Graphs

Let G = (V (G), E(G)) be a simple undirected graph of order n, and let S subset of (G). If every vertex in V (G) - S is adjacent to at least one vertex in S, then the set S is called a dominating set. The domination number of G is the minimum cardinality taken over all sets of S, and it is denoted by gamma(G). Recently, the effect of one or more edges deletion on the domination number has been examined in many papers. Let F subset of E(G). The bondage number b(G) of G is the minimum cardinality taken over all sets of F such that gamma(G - F) > gamma(G). In the literature, a lot of domination and bondage parameters have been defined depending on different properties. In this paper, we investigate the bondage, strong and weak bondage numbers of complementary prism graphs of some well-known graph families

Combining the Concepts of Residual and Domination in Graphs

###EgeUn###Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by gamma(R)(uk)(G), is the minimum cardinality of dominating set in G that received from the graph G where the vertex v(k) and all links of the vertex v(k) are deleted. The residual domination number of graphs G is defined as gamma(R)(G) = minv(k)is an element of V(G){gamma(R)(vk)(G)} . The average lower residual domination number of G is de- fined by gamma(R)(av)(G) = 1/vertical bar V(G)vertical bar Sigma(vk is an element of V(G)) gamma(R)(vk)(G). In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families

VERTEX VULNERABILITY PARAMETER OF GEAR GRAPHS

WOS: 000293753200011For a vertex v of a graph G = (V, E), the independent domination number (also called the lower independence number) i(v) (G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is i(av) (G) = 1 /|V(G)| Sigma(v epsilon V(G)) i(v) (G). In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offere

Porous exponential domination number of some graphs

WOS: 000584048900001Let G be a graph and S subset of V(G). If n-ary sumation u is an element of S12d(u,v)-1 >= 1 for all v is an element of V(G), then S is a porous exponential dominating set for G, where d(u, v) is the distance between vertices u and v. the smallest cardinality of a porous exponential dominating set is the porous exponential domination number of G and is denoted by gamma e*(G). in this article, we examine porous exponential domination number of some shadow graphs and trees such as comet, double comet, double star, binomial tree, and generalized caterpillar graphs

SOME RESULTS FOR THE RUPTURE DEGREE

WOS: 000335513900009The rupture degree of an incomplete connected graph G is defined by r(G) = max{w(G - S) - vertical bar S vertical bar - m(G - S) : S CV(G), w(G - S) >= 2}, where w(G - 5) denotes the number of components in the graph G - S and m(G - 5) is the order of the largest component of G - S. This parameter can be used to measure the vulnerability of a graph. In this paper, some bounds consisted of the relationships between the rupture degree and some vulnerability parameters on the rupture degree of a graph are given