Related papers: Spanning tree packing, edge-connectivity and eigen…

A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for $n$ and $C$ large enough, if $G$ is an edge-colored copy of $K_n$ in which each color…

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a…

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an…

An edge subset $S$ of a connected graph $G$ is called an anti-Kekul\'{e} set if $G-S$ is connected and has no perfect matching. We can see that a connected graph $G$ has no anti-Kekul\'{e} set if and only if each spanning tree of $G$ has a…

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree…

We prove that if $\Gamma$ is a connected graph with minimum degree $\delta$ and Laplacian eigenvalues $0=\mu_1<\mu_2\leqslant \cdots \leqslant \mu_n$, then the toughness of $\Gamma$ is bounded below by $\mu_2/(\mu_n-\delta)$.

Let $F(G)$ be the number of spanning forests in a graph $G$ and $\mathcal{C}(n,d)$ be the set of all connected $d$-regular simple graphs of order $n$. Define $\widehat{f}_{d}=\liminf_{n\rightarrow \infty}\{F(G)^{1/n}:G\in…

For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of…

In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the…

For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every…

The algebraic connectivity $a(G)$, defined as the second smallest eigenvalue of the Laplacian matrix $L(G)$, admits a well-known variational characterization involving the minimization of a quadratic form subject to an $\ell_{2}$-norm…

For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity} between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of $G$, denoted…

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this…

For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The $\theta_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $\theta_r$. This…

The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees…

For a graph $G$ and for two distinct vertices $u$ and $v$, let $\kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written…

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…

The tree breadth ${\rm tb}(G)$ of a connected graph $G$ is the smallest non-negative integer $\rho$ such that $G$ has a tree decomposition whose bags all have radius at most $\rho$. We show that, given a connected graph $G$ of order $n$ and…