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

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $δ$. 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 $λ_i$ and $μ_i$ 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 $i = 1, 2, \ldots, n$, respectively. In this paper, we investigate the relationship between Laplacian eigenvalues and property $P(k, δ)$. 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 $V_1, V_2, \ldots, V_{t+1}$ satisfying $V(G) \setminus \bigcup_{i=1}^{t+1} V_i \neq \emptyset$ and edge connectivity $κ'(G) = e(V_i, V(G) \setminus V_i)$ for any $i = 1, 2, \ldots, t+1$. For the class of graphs $\mathcal{G}_1$ with minimum degree $δ$, we provide a sufficient condition involving the third smallest Laplacian eigenvalue $μ_{n-2}(G)$ for a graph $G\in \mathcal{G}_1$ to have property $P(k, δ)$. Similarly, for the class of graphs $\mathcal{G}_2$ with minimum degree $δ$, we establish a corresponding sufficient condition involving the fourth smallest Laplacian eigenvalue $μ_{n-3}(G)$ for a graph $G\in \mathcal{G}_2$ to have property $P(k, δ)$. Furthermore, we extend the spectral conditions for all the results about $μ_{n-2}(G)$, $μ_{n-3}(G)$ and $λ_2(G)$ to the general graph matrices $aD(G) + A(G)$ and $aD(G) + bA(G)$.

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