Related papers: A characterization on trees $T$ with $m(T, \lambda…

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two…

An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\Psi(G)$. In this paper, an algorithm to count…

Let $G$ be a connected graph of order $n$ with domination number $\gamma(G)$. Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue $\lambda$ of $G$ with multiplicity…

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number…

We study higher-dimensional analogues of graph-theoretic trees within the class of pure n-simplicial complexes. Focusing on the case m = n-1 in Dewdney's (m, n)-tree framework, we introduce refined notions of path and circuit sequences that…

Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one of the Laplacian eigenvalue having multiplicity $k$. It is well known that $\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined by Das,…

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also…

For a finite simple undirected graph $G$, let $\gamma(G)$ denote the size of a smallest dominating set of $G$ and $\mu(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities.…

Given a graph $G$ rooted at a vertex $r$ and weight functions, $\gamma, \tau: E(G) \rightarrow \mathbb{R}$, the generalized cable-trench problem (CTP) is to find a single spanning tree that simultaneously minimizes the sum of the total edge…

The tree-depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree-depth of random graphs. For dense graphs, p>> 1/n, the tree-depth of a random graph G is a.a.s.…

We show that every connected graph $G$ has a tree decomposition indexed by a tree $T$ such that $T$ is a subgraph of $G$ and the width of the tree decomposition is bounded from above by a function of the pathwidth of $G$. This answers a…

We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$\alpha$ if and only if it is "$(\mathrm{tw},\omega)$-bounded" (that is, for all $t\in \mathbb N$, the class of all…

The study of eigenvalue multiplicities plays a central role in the spectral theory of signed graphs, extending several classical results from the unsigned setting. While most existing work focuses on the nullity of a signed graph (the…

The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \cdots, T_k$ connecting…

In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_{\alpha}(G)$ as $$A_{\alpha}(G)=\alpha…

For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $\tau_G(M)$ is obtained,…

A signed tree-coloring of a signed graph $(G,\sigma)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,\sigma)$ whose vertex set is the set of…

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick,…