Malaysian Journal of Mathematical Sciences, May 2020, Vol. 14, No. 2

More on the Sixth Coefficient of the Matching Polynomial in Regular Graphs

Alikhani, S. and Soltani, N.

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Received date: 27 January 2019
Accepted date: 1 April 2020

A matching set \(M\) in a graph \(G\) is a collection of edges of \(G\) such that no two edges from \(M\) share a vertex. The matching polynomial of \(G\) of order \(n\) is defined by \(\mu(G,x)=\sum_{r=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^{r}\rho(G,x)x^{n-2r}\) where \(\rho(G,x)\) is the number of matching sets of \(G\) with cardinality \(r\) and \(p(G,0)\) is considered to be one. A graph that is characterized by its matching polynomial is said to be matching unique. In this paper, we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order 10 is matching unique.

Keywords: Saturation number, matching polynomial and regular graph



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