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Multipath zeta functions of graphs. (English) Zbl 0988.11040

Hejhal, Dennis A. (ed.) et al., Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15-26, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 109, 601-615 (1999).
In H. M. Stark and A. A. Terras, Zeta-functions of finite graphs and coverings. Part I [Adv. Math. 121, 124–165 (1996; Zbl 0874.11064)] we investigated three sorts of zeta functions (vertex, edge, and path) that may be attached to connected irregular graphs \(X\). These may be viewed as analogues of Dedekind zeta functions of a number field. The vertex zeta function is also known as the Ihara-Selberg zeta function and has been much studied. When the graph is regular the Riemann hypothesis for the vertex or Ihara-Selberg zeta function is equivalent to the graph being Ramanujan in the sense of A. Lubotzky, R. S. Phillips and P. Sarnak [Combinatorica 8, 261–277 (1988; Zbl 0661.05035)]. Recently X.-S. Lin and Z. Wang [Knots, braids, and mapping class groups – papers dedicated to Joan S. Birman, New York 1998, AMS/IP Stud. Adv. Math. 24, 107–121 (2001; Zbl 0992.57001)] have connected the Ihara-Selberg zeta function with the Alexander polynomial of a knot.
Many examples in the Advances paper mentioned above suggested that these graph zeta functions have factorizations similar to those for Dedekind zeta functions of algebraic number fields as products of Artin \(L\)-functions. To do this, one needs a Galois theory of normal unramified coverings of finite graphs. The paper under review begins this theory with the case of quadratic coverings; i.e., coverings with 2 sheets. H. M. Stark and A. A. Terras, “Zeta functions of finite graphs and coverings”. Part II [Adv. Math. 154, 132–195 (2000; Zbl 0972.11086)] give a discussion of Galois theory and Artin L-functions of the 3 types for arbitrary finite normal unramified coverings of finite irregular graphs which may have multiple edges and loops. In this framework a prime becomes a prime cycle \([C]\) in a graph, meaning roughly that \(C\) is a closed path with no backtracking and you only go around the cycle once. The equivalence class \([C]\) consists of the same path with different starting vertices.
A second component of the paper under review is an algorithm that shows how to specialize path zeta and \(L\)-functions to edge zeta and \(L\)-functions. The edge zetas are specialized to vertex or Ihara-Selberg zetas by setting all the variables equal to one complex variable. So, in some sense the path \(L\)-function is one \(L\)-function to rule them all. The specialization algorithm described here is an improvement over that of Part I [loc. cit.].
The path \(L\)-function is defined using the fact that the fundamental group of \(X\) can be identified with the free group generated by edges of \(X\) left out of a spanning tree. Call these oriented edges \(e_{1},\dots ,e_{r},e_{1} ^{-1},\dots ,e_{r}^{-1}.\) Create the \(2r\times 2r\) path matrix \(Z\) with \(ij\) entry the complex variable \(z_{ij}\) if \(e_{j}\neq e_{i}^{-1}\) and \(z_{ij}=0\) otherwise. We also define \(z(e_{i},e_{j})=z_{ij}.\)
Consider the prime cycle \([C]\) of \(X\) with \(C=a_{1}\cdots a_{s},\) where \[ a_{j}\in\left\{ e_{1},\dots ,e_{r},e_{1}^{-1},\dots ,e_{r}^{-1}\right\} \] and \(C\) is a reduced product of generators of the fundamental group. Here “reduced” means that \(a_{j+1}\neq a_{j}^{-1},\) for all \(j\), and \(a_{s}\neq a_{1}^{-1}.\) One can identify \([C]\) with a primitive reduced conjugacy class in the fundamental group of \(X\). The path norm of the reduced prime \(C=a_{1}\cdots a_{s}\) is \(N_{P}(C)=z(a_{s},a_{1})\prod_{i=1} ^{s-1}z(a_{i},a_{i+1}).\) The path \(L\)-function for a quadratic cover \(Y/X\) is defined by \[ L_{P}(Z,\rho)=\prod_{{[ C ]\prime\in X} }\det\left( 1-\chi(C)N_{P}(C)\right) ^{-1}. \] Here \(\chi(C)=1\) if \(C\) lifts to a closed path in \(Y\) and \(\chi(C)=-1\) otherwise. The paper under review explains how to specialize the \(Z\)-variables of the graph \(Y\) to obtain a factorization of the path zeta function of \(Y\) as a product of the path zeta function of \(X\) with \(L_{P}(Z,\rho)\). As usual in this subject, the inverse of \(L_{P}(Z,\rho)\) is a polynomial. There is also an example in which the factorization leads to an identity for a \(6\times 6\) determinant of a matrix with many variables as a product of two \(4\times 4\) determinants.
Another reference for a quick introduction to the subject is H. M. Stark and A. A. Terras [Dynamical, spectral, and arithmetic zeta functions, San Antonio, TX, 1999, Contemp. Math. 290, 181–195 (2001; Zbl 1019.11024)].
For the entire collection see [Zbl 0919.00047].

MSC:

11M41 Other Dirichlet series and zeta functions
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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