The coefficients of the reduced Bartholdi zeta function

In this paper, we establish a new zeta function based on the Bartholdi zeta function for an undirected graph G called the reduced Bartholdi zeta function. We study the relation between its coefficients and the structure of the graph, and demonstrate that the coefficients count the star subgraphs in the symmetric digraph D(G). Moreover, we investigate the properties of semi-principle minors extracted from the adjacency matrix of the oriented line graph of G. We also present a general formula for calculating all the coefficients of the reduced Bartholdi zeta function.