Abstract
Let G be a finite connected graph with vertex set V and (undirected) edge set E. Other authors [1,2,3,4,6] have previously introduced zeta functions attached to G in one and several variables. These all turned out to be the inverse of polynomials in the variables. The variables were associated to vertices and to edges of G. In [5], Stark and Terras introduced a multipath zeta function with variables associated to pairs of generators of the fundamental group of G. This zeta function depends only upon the rank of G (the rank of the fundamental group of G). Nevertheless, it was shown in [5] that the path variables could be specialized as monomials in the edge or vertex variables so as to yield all previously defined zeta functions of graphs.
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References
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© 1999 Springer Science+Business Media New York
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Stark, H.M. (1999). Multipath Zeta Functions of Graphs. In: Hejhal, D.A., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds) Emerging Applications of Number Theory. The IMA Volumes in Mathematics and its Applications, vol 109. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1544-8_26
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DOI: https://doi.org/10.1007/978-1-4612-1544-8_26
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