Some Results Of The Matching And Matching Energy

Gutman (Topological properties of benzenoid systems, Theort. Chim.Acta,45(1977),307-315.) proved the following result:For any hexagonal chain H, there exits a corresponding caterpillar tree T(H), such that the number of perfect matchings of H is equal to the number of matchings of T(H).In the second chapter of this paper, we obtained a similar result:For any polyomino chain Q, there exists a corresponding caterpillar tree C(Q), such that the number of perfect matchings of Q is equal to the number of matchings of C(Q).Gutman and Wagner (The matching energy of a graph, Discr. Appl. Math,160(2012),2177-2187.) defined the matching energy, they also give some basic properties and asymptotic results of matching energy.In the third chapter of this paper, we study the upper bound of matching energy of a connected graph with connectivity, chromatic number, characterize the connected graph G of order n with the connectivity κ(or chromatic number χ) which has the maximum matching energy.