The Ordering of the Unicyclic Graphs with respect to Largest Matching Root with Given Matching Number

Abstract

The matching roots of a simple connected graph G are the roots of the matching polynomial which is defined as M G x = ∑ k = 0 n / 2 − 1 k m G , k x n − 2 k , where m G , k is the number of the k matchings of G . Let λ 1 G denote the largest matching root of the graph G . In this paper, among the unicyclic graphs of order n , we present the ordering of the unicyclic graphs with matching number 2 according to the λ 1 G values for n ≥ 11 and also determine the graphs with the first and second largest λ 1 G values with matching number 3.