New York Journal of Mathematics
Volume 17 (2011) 295-320


Terrence Bisson and Aristide Tsemo

Homotopy equivalence of isospectral graphs

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Published: June 3, 2011
Keywords: category of directed graphs, topos, Quillen model structure, homotopy category, cycles, algebraic graph theory, zeta function
Subject: 05C20,18G55, 55U35.

In previous work we defined a Quillen model structure, determined by cycles, on the category Gph of directed graphs. In this paper we give a complete description of the homotopy category of graphs associated to our model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Z-sets, and to show that two finite directed graphs are almost-isospectral if and only if they are homotopy-equivalent in our sense.

Author information

Terrence Bisson:
Department of Mathematics and Statistics, Canisius College, Buffalo NY, 14208, USA

Aristide Tsemo:
102 Goodwood Park, Apt. 614, Toronto, Ontario M4C 2G8, Canada