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 $M_G\left(x\right)={\displaystyle {\sum }_{k=0}^{n/2}{\left(-1\right)}^{k}m\left(G,k\right)x^{n-2k}}$ , where $m\left(G,k\right)$ is the number of the $k$ matchings of $G$ . Let ${\lambda }_1\left(G\right)$ 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 ${\lambda }_1\left(G\right)$ values for $n\ge 11$ and also determine the graphs with the first and second largest ${\lambda }_1\left(G\right)$ values with matching number 3.