Zeta functions of graphs and coverings

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We express an L-function of a regular covering of a graph G by using that of G. Moreover, we treat zeta functions and L-functions of a semiregular bipartite graph, its line graph, its middle graph and their regular coverings, and present an analogue of the Selberg trace formula for an L-function of a semiregular bipartite graph.We consider weighted zeta functions and L-functions of digraphs, and give their determinant expressions. Moreover, we present determinant expressions for weighted zeta functions and weighted L-functions of graphs, and express the weighted zeta function of a regular covering of a graph as a product of its weighted L-functions. By using a similar method to the above one, we present determinant expressions for the weighted complexities of a graph and its (regular or irregular) covering. Furthermore, we present a new decomposition formula for the weighted zeta function of a (regular or irregular) covering of a graph, and study the structure of a balanced covering of a unbalanced graph as an application.Finally, we consider the Bartholdi zeta functions and Bartholdi L-functions of a graph, a digraph and their covering, and present their determinant expressions and decomposition formulas. Furthermore, we present decomposition formulas for the Bartholdi zeta function of some branched covering of a graph, and the weighted Bartholdi zeta function of a graph.