Some graphs with small second eigenvalue

Abstract

For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0Firpepe0de9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdadaGcaa% qaaiaadchaaSqabaaaaa!373F! $$2\sqrt p $$ with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp k nodes of degreep of second eigenvalue % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0Firpepe0de9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgsMiJkaacI% cacaWGRbGaeyOeI0IaaGymaiaacMcadaGcaaqaaiaadchaaSqabaaa% aa!3C29! $$ \leqslant (k - 1)\sqrt p $$ .