Graphs with integer matching polynomial zeros

In this paper, we study graphs whose matching polynomials have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs. We show that apart from K7∖(E(C3)∪E(C4)) there is no connected k-regular matching integral graph if k≥2. It is also shown that if G is a graph with a perfect matching, then its matching polynomial has a zero in the interval (0,1]. Finally, we describe all claw-free matching integral graphs.