Algebraic connectivity of Kronecker products of line graphs

TL;DR

This paper characterizes trees whose line graph Kronecker products with complete graphs have specific algebraic connectivity properties, including conditions for Laplacian integrality and special cases involving trees of diameter four.

Contribution

It provides a complete characterization of trees for which the algebraic connectivity of their line graph Kronecker product with K_m equals m-1, and explores Laplacian integrality conditions.

Findings

Characterization of trees with algebraic connectivity of L(X)×K_m equal to m-1.

Necessary and sufficient conditions for Laplacian integrality of L(X)×K_m.

Analysis of algebraic connectivity for trees of diameter 4 and k-book graphs.

Abstract

Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)\times K_m$ is equal to $m-1$, where $\times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)\times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)\times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.