From Integer Sequences to Block Designs via Counting Walks in Graphs

TL;DR

This paper links integer sequences derived from counting walks in graphs to the structure of block designs, showing that certain graphs with specific walk sequences are incidence graphs of block designs and possess regularity and Ramanujan properties.

Contribution

It establishes a novel connection between integer sequences from graph walks and the construction of block designs via Kronecker products, identifying unique graphs with specific walk properties.

Findings

Sequences correspond to counts of odd and even walks in complete graphs.

Identifies a unique family of graphs with matching walk sequences.

These graphs are incidence graphs of block designs, distance-regular, and Ramanujan.

Abstract

We define numbers of the type Oj(N) and Ej(N) and the corresponding integer sequences. We prove that these integer sequences, e.g., SO(N) and SE(N) correspond to the number of odd and even walks in complete graphs. We then prove that there is a unique family of graphs which have exactly the same sequence of odd walks between connected nodes and of even walks between pairs of nodes at distance two, respectively. These graphs are obtained as the Kronecker product. We show that they are the incidence graphs of block designs, are distance-regular and Ramanujan graphs.