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{\bf Yaim Cooper}
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{\bf Properties Determined by the Ihara Zeta Function of a Graph}
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In this paper, we show how to determine several properties of a finite
graph $G$ from its Ihara zeta function $Z_{G}(u)$. If $G$ is connected
and has minimal degree at least 2, we show how to calculate the number
of vertices of $G$. To do so we use a result of Bass, and in the case
that $G$ is nonbipartite, we give an elementary proof of Bass'
result. We further show how to determine whether $G$ is regular, and
if so, its regularity and spectrum. On the other hand, we extend work
of Czarneski to give several infinite families of pairs of
non-isomorphic non-regular graphs with the same Ihara zeta
function. These examples demonstrate that several properties of
graphs, including vertex and component numbers, are not determined by
the Ihara zeta function. We end with Hashimoto's edge matrix T. We
show that any graph $G$ with no isolated vertices can be recovered
from its $T$ matrix. Since graphs with the same Ihara zeta function
are exactly those with isospectral T matrices, this relates again to
the question of what information about $G$ can be recovered from its
Ihara zeta function.

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