The average lower reinforcement number of a graph

¹ Department of Mathematics, Faculty of Science, Karabük University, 78050, Karabük, Turkey.
tufanturaci@karabuk.edu.tr
² Turgutlu Vocational Training School, Celal Bayar University, 45400 Manisa, Turkey.
ersin.aslan@cbu.edu.tr

Received: 22 February 2016
Accepted: 16 June 2016

Abstract

Let G = (V(G),E(G)) be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. The lower reinforcement number, denoted by r_e^∗(G), is the minimum cardinality of reinforcement set in G that contains the edge e^∗ of the complement graph G̅ . The average lower reinforcement number of G is defined by r_av(G)=1/|E(G̅)| ∑_{e** ∈ E(G̅)} r_e*(G) .In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families.