On the $ P_3 $-hull numbers of $ q $-Kneser graphs and Grassmann graphs

TL;DR

This paper calculates the minimum initial infected set size needed to eventually infect entire $q$-Kneser and Grassmann graphs through a specific infection process, advancing understanding of spreading dynamics in these combinatorial structures.

Contribution

It provides explicit computations of the $P_3$-hull numbers for $q$-Kneser and Grassmann graphs, a novel contribution to the study of infection spreading in these graphs.

Findings

Computed the $P_3$-hull numbers for $q$-Kneser graphs.

Computed the $P_3$-hull numbers for Grassmann graphs.

Enhanced understanding of infection dynamics in combinatorial graph structures.

Abstract

Let $S$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, where $q$ is necessarily a prime power. Denote $K_q(n,k)$ (resp. $J_q(n,k)$) to be the \emph{$q$-Kneser graph} (resp. \emph{Grassmann graph}) for $k\geq 1$ whose vertices are the $k$-dimensional subspaces of $S$ and two vertices $v_1$ and $v_2$ are adjacent if $\dim(v_1\cap v_2)=0$ (resp. $\dim(v_1\cap v_2)=k-1$). We consider the infection spreading in the $ q $-Kneser graphs and the Grassmann graphs: a vertex gets infected if it has at least two infected neighbors. In this paper, we compute the $ P_3 $-hull numbers of $K_q(n,k)$ and $J_q(n,k)$ respectively, which is the minimum size of a vertex set that eventually infects the whole graph.