Spectral Properties of 3-Regular Graphs Related to One-Face Maps

Abstract

We study cycle-toothpick graphs, a class of graphs related to one-face maps which were first introduced by Flaschka. We explore the relationship between the spectrum of the adjacency matrix of a cycle-toothpick graph and the genus of the associated one-face map. In particular we focus on the eigenvalue distributions of sequences of cycle-toothpick graphs by using a theorem of McKay in [16]. We prove that sequences of cycle-toothpick graphs with bounded genus will not satisfy the assumption in McKay's theorem. Through application of existing results we show that simple cycle-toothpick graphs on 2n vertices sampled uniformly at random are likely to have genus close to n/2 -ln n /2 and desirable spectral properties. In order to better understand sequences of cycle-toothpick graphs with unbounded genus we construct cycle-toothpick graphs via lifts of graphs. Using particular lifts of graphs called permuted toothpick lifts, we are able to construct cycle-toothpick graphs of maximal genus that are growing in size. We show that such graphs will not satisfy the assumptions in McKay's theorem. Moreover we provide numerical results that such graphs are unlikely to satisfy the Ramanujan condition. We also use permuted toothpick lifts to construct cycle-toothpick graphs that appear to have genus n/2 -ln n/2 on average but are unlikely to be constructed by sampling a random cycle-toothpick graph. We see numerically that such graphs will satisfy the Ramanujan condition. We show that the eigenvalue distribution of sequences of such graphs will converge to that of the tree. Together these results suggest that cycle-toothpick graphs on 2n vertices with genus n/2 - In n /2 are likely to satisfy the Ramanujan condition and sequences of such graphs are likely to have an eigenvalue distribution that converges to that of the infinite 3-regular tree.