Laplacian eigenvalue conditions for edge-disjoint spanning trees and a forest with constraints

Abstract

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\hat{I}´$. A graph $G$ is said to have property $P(k,d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)|>\frac{d-1}{d}(|V(G)|-1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the degree diagonal matrix of $G$. We denote $\hat{I}{»}_i$ and $\hat{I}{¼}_i$ as the $i$th largest eigenvalue of the adjacency matrix $A(G)$ of $G$ and the Laplacian matrix $L(G)=D(G)-A(G)$ of $G$ for $i=1,2,\dots ,n$, respectively. In this paper, we investigate the relationship between Laplacian eigenvalues and property $P(k,\hat{I}´)$. Let $t$ be a positive integer, and define ${\mathcal{G}}_t$ as the set of simple graphs such that each $G\in {\mathcal{G}}_t$ contains at least $t+1$ non-empty disjoint proper subsets $V_1,V_2,\dots ,V_{t+1}$ satisfying $V(G)\setminus {\bigcup }_{i=1}^{t+1}{V}_i\ne \varnothing$ and edge connectivity $\hat{I}{º}^{\prime }(G)=e(V_i,V(G)\setminus V_i)$ for any $i=1,2,\dots ,t+1$. For the class of graphs ${\mathcal{G}}_1$ with minimum degree $\hat{I}´$, we provide a sufficient condition involving the third smallest Laplacian eigenvalue $\hat{I}{¼}_{n-2}(G)$ for a graph $G\in {\mathcal{G}}_1$ to have property $P(k,\hat{I}´)$. Similarly, for the class of graphs ${\mathcal{G}}_2$ with minimum degree $\hat{I}´$, we establish a corresponding sufficient condition involving the fourth smallest Laplacian eigenvalue $\hat{I}{¼}_{n-3}(G)$ for a graph $G\in {\mathcal{G}}_2$ to have property $P(k,\hat{I}´)$. Furthermore, we extend the spectral conditions for all the results about $\hat{I}{¼}_{n-2}(G)$, $\hat{I}{¼}_{n-3}(G)$ and $\hat{I}{»}_2(G)$ to the general graph matrices $aD(G)+A(G)$ and $aD(G)+bA(G)$.