  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.
