More graphs with pair state transfer

Abstract

This paper has two main goals. First, we characterize perfect state transfer between $s$-pair states in strongly regular graphs, as well as graphs in association schemes admitting perfect state transfer between vertices. The second goal is to provide a unified approach for constructing non-regular graphs admitting pair state transfer$-$relative to the adjacency, Laplacian, and signless Laplacian matrix$-$between the same pair of states at the same time. In particular, we show that for each $k\ge 5$, there are infinitely many connected graphs with maximum valency $k$ admitting this property. We also utilize graph products to generate new infinite families of graphs with pair state transfer.