We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-ba...

We consider strongly regular graphs r = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman-Singleton graph, and the triangular graphs T(q) with q = 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We...

The spectral radius of connected non-regular graphs is considered. Let λ1 be the largest eigenvalue of the adjacency matrix of a graph G on n vertices with maximum degree ∆. By studying the λ1-extremal graphs, it is proved that if G is non-regular and connected, then ∆− λ1 > ∆+ 1 n(3n+∆− 8) . This improves the recent results by B.L. Liu et al. AMS subject classifications. 05C50, 15A48.

A Neumaier graph is a non-complete edge-regular containing regular clique. that not strongly called strictly graph. In this work we present new construction of graphs, and using Jacobi sums, show our produces infinitely many instances. Moreover, prove some necessary conditions for the existence (strictly) graphs allow us to several parameter sets are admissible.

We describe for the first time how the 5-regular simple planar graphs can all be obtained from a simple family of starting graphs by repeatedly applying a few expansion operations. The proof uses an innovative amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic 5-regular planar graphs ...