Graphs with Integer Matching Polynomial Roots
  In this paper, we study graphs whose matching polynomial 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 n (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. 
S. Akbari, P. Csikvari, A. Ghafari, S. Khalashi Ghezelahmad, M. Nahvi
math.CO cs.DM