The Derivative Of A Function Related To The Zeta Function Of Some Graphs

Suppose that G is a graph with n vertices and m edges, the Zeta function of G could be expressed ZG（u）= （1-u²）n-m/f（u）, where f（u）= det（I_n-uA（G）+u²（D（G）-I_n））, A（G） and D（G） are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. In this paper, we study the derivatives of a function f（u）at u=±1 related to the Zeta function of a graph G. At first, we obtain the first derivative at-1 and the second derivatives at±1 of f（u） of an r-regular graph G that are f’（-1）, f"（1） and f’’（-1） could be expressed by the number of vertices n, the number of edges m, the number of spanning trees k（G）, the kirchhoff index Kf（G） and the weights of the TU-subgraphs ω of G, respectively. We also obtain the corresponding derivatives of a （d₁,d₂）-semiregular graph G could be expressed by the number of vertices n, the number of edges m, the number of spanning trees κ（G）, the kirchhoff index Kf（G）, the Degree Kirchhoff index Kf*（G） and the degree resistance distance Dr（G） of G, respectively.