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The coefficients of the reduced Bartholdi zeta function. (English) Zbl 1346.05178

Summary: 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 \(\mathcal{D}(G)\). 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.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
15A15 Determinants, permanents, traces, other special matrix functions
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References:

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