For a median graph G and a vertex v of G that is not a cut-vertex we show that G − v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whos...

Let $G$ be a finite group. Define graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to adjacent if and only $\langle x,y\rangle$ is cyclic. Denote this $\Delta(G)$. The $\Delta(G)$ has appeared in literature under names cyclic deleted enhanced power graph. If $H$ are nontrivial groups, then $\Delta(G\times H)$ completely characterized. In particular,...

The analysis of vulnerability in networks generally involves some questions about how the underlying graph is connected. One is naturally interested in studying the types of disruption in the network that maybe caused by failures of certain links or nodes. In terms of a graph, the concept of connectedness is used in dierent forms to study many of the measures of vulnerability. When certain vert...

Brinkmann graph is a 4-regular with 21 nodes and 42 edges discovered by Gunnar in 1992. To our knowledge, the research specifically on still hard to find. Therefore, this was carried out analyze term of its centrality. The centrality measures used are degree, betweenness, closeness In paper, we presented not only but also after node deletion see how impact graph. Before deletion, results showed...

We consider the Critical Node Problem: given an undirected graph and an integer number K, at most K nodes have to be deleted from the graph in order to minimize a connectivity measure in the residual graph. We combine the basic steps used in common greedy algorithms with some flavour of local search, in order to obtain simple “hybrid” heuristic algorithm. The obtained algorithms are shown to be...

in this paper, we present a binary-linear optimization model to prevent the spread of an infectious disease in a community. the model is based on the remotion of some connections in a contact network in order to separate infected nodes from the others. by using this model we nd an exact optimal solution and determine not only the minimum number of deleted links but also their exact positions. ...

The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and...

In this paper, we present a binary-linear optimization model to prevent the spread of an infectious disease in a community. The model is based on the remotion of some connections in a contact network in order to separate infected nodes from the others. By using this model we nd an exact optimal solution and determine not only the minimum number of deleted links but also their exact positions. T...