Abstract
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.
Highlights
Common-Neighbourhood and Other Measures on GraphsOther measures provide bounds on 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
While the ordinary connectivity is the minimum number of vertices whose removal separates the graph into at least one connected pair of vertices and a isolated vertex, the average connectivity is a measure for the expected number of vertices that have to be removed to separate a randomly chosen pair of vertices
Summary
Other measures provide bounds on the common-neighbourhood of a graph. We give some theorems relating to common-neighbourhood and graph parameters. For any non-regular graph G, ∆(G) denotes maximum vertex degree and δ(G) denotes minimum vertex degree of the graph G. For all (u,v ) pairs of the maximum independent set of G, d (u,v ) ≥ 2. Let G1 and G2 be graphs with n vertices .If σ(G1) < σ(G2), N (G1) > N (G2). Proof: Let S 1and S 2 be the minimum vertex dominating sets of graphs G1 and G2, respectively. U∈S1 N (u, v) > n−1 u∗∈S2 N (u∗, v) n−1 and by the definition of common-neighbourhood, N (G1) > N (G2)
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