Max-cut and extendability of matchings in distance-regular graphs

A connected graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. Since its introduction by Plummer in the 1980s, this combinatorial parameter has been studied for many classes of interesting graphs. In 2005, Brouwer and Haemers proved that every distance-regular graph of even order is $1$-extendable and in 2014, Cioab? and Li showed that any connected strongly regular graph of even order is 3-extendable except for a small number of exceptions.In this paper, we extend and generalize these results. We prove that all distance-regular graphs with diameter $D\ge 3$ are 2-extendable and we also obtain several better lower bounds for the extendability of distance-regular graphs of valency $k\ge 3$ that depend on $k$, $\lambda$ and $\mu$, where $\lambda$ is the number of common neighbors of any two adjacent vertices and $\mu$ is the number of common neighbors of any two vertices in distance two. In many situations, we show that the extendability of a distance-regular graph with valency $k$ grows linearly in $k$. We conjecture that the extendability of a distance-regular graph of even order and valency $k$ is at least $?k/2??1$ and we prove this fact for bipartite distance-regular graphs.In course of this investigation, we obtain some new bounds for the max-cut and the independence number of distance-regular graphs in terms of their size and odd girth and we prove that our inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.