Induced forests in some distance-regular graphs

TL;DR

This paper investigates the size and structure of the largest induced forests in certain distance-regular graphs, introducing bounds and special types of forests, and providing examples where these bounds are tight or can be exceeded.

Contribution

It introduces a variation of the ratio bound for induced forests and defines canonical induced forests, with examples in distance-regular graphs where these bounds are achieved.

Findings

Derived an upper bound on the size of induced forests.

Identified conditions where canonical forests are maximal.

Provided examples exceeding canonical forest sizes.

Abstract

In this article, we study the order and structure of the largest induced forests in some families of graphs. First we prove a variation of the ratio bound that gives an upper bound on the order of the largest induced forest in a graph. Next we define a \textsl{canonical induced forest} to be a forest that is formed by adding a vertex to a coclique and give several examples of graphs where the maximal forest is a canonical induced forest. These examples are all distance-regular graphs with the property that the Delsarte-Hoffman ratio bound for cocliques holds with equality. We conclude with some examples of related graphs where there are induced forests that are larger than a canonical forest.