Algebraic connectivity of Kronecker products of line graphs

TL;DR

This work characterizes when the algebraic connectivity of the Kronecker product $L(X)\times K_m$ attains the value $m-1$ for trees $X$, by linking the spectrum of $L(X\times K_m)$ to the spectra of $(m-1)L(X)$ and $Q_{m-1}(X)$. It shows that for $X$ in the family $\mathfrak{S}_m$, the equality $a(β_m(X))=m-1$ occurs if and only if $X$ is a $T(1,s,t)$ with $s,t\ge2$, and provides necessary and sufficient conditions for $β_m(X)$ to be Laplacian integral. The paper then analyzes algebraic connectivity for several graph classes (windmill graphs $W(\eta,\mu)$, their variants $W'(\eta,\mu)$, and $k$-book graphs) under the same product, yielding explicit eigenvalue formulas and conditions for the ${a}$-values. These results deepen understanding of how line-graph structure and graph products influence Laplacian spectra and related extremal connectivity properties, with implications for spectral graph theory and network design.

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.

Figures (6)

• Figure 1: $P_4,L(P_4),\beta_3(P_4),\beta_4(P_4)$ and $\beta_5(P_4)$
• Figure 2: $T(2,3,2),L(T(2,3,2))$ and $\beta_2(T(2,3,2))$
• Figure 3: $T(1,3,2),L(T(1,3,2))$
• Figure 4:
• Figure 5: $W(2,3),W^\prime(3,3),W(2,3)\times K_2$ and $W^\prime(3,3)\times K_2$

Theorems & Definitions (23)

• Example 1
• Proposition 2
• Theorem 3
• Example 4
• Lemma 5
• Theorem 6
• proof
• Corollary 7
• proof
• Definition 8