A generalized characteristic polynomial of a graph having a semifree action. (English) Zbl 1130.05036
Summary: For an abelian group \(\varGamma \), a formula to compute the characteristic polynomial of a \(\varGamma \)-graph has been obtained by J. Lee and H. K. Kim [Linear Algebra Appl. 307, 35–46 (2000; Zbl 0991.05072)]. As a continuation of this work, we give a computational formula for the generalized characteristic polynomial of a \(\varGamma \)-graph when \(\varGamma \) is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Citations:
Zbl 0991.05072References:
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