2017/08/31 by Kloster, Kyle, Král', Daniel, Sullivan, Blair D.
#05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.09700
A graph is said to be walk-regular if, for each ℓ ≥ 1, every vertex is contained in the same number of closed walks of length ℓ. We construct a 24-vertex graph H4 that is not walk-regular yet has maximized walk entropy, SV(H4,β) = log 24, for some β>0. This graph is a counterexample to a conjecture of Benzi [Linear Algebra Appl.~443 (2014), 395--399, Conjecture 3.1]. We also show that there exist infinitely many temperatures β0>0 so that SV(G,β0)=log nG if and only if a graph G is walk-regular.