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

An $\alpha$-thin tree $T$ of a graph $G$ is a spanning tree such that every cut of $G$ has at most an $\alpha$ proportion of its edges in $T$. The Thin Tree Conjecture proposes that there exists a function $f$ such that for any $\alpha >…

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number…

The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted…

If $G$ is a graph and $\mathbf{m}$ is an ordered multiplicity list which is realizable by at least one symmetric matrix with graph $G$, what can we say about the eigenvalues of all such realizing matrices for $\mathbf{m}$? It has sometimes…

Let $G$ be a connected (non-complete) $d$-regular graph with $d\geq3$. Let $c(G-S)$ denote the number of components of $G-S$ for any cut $S$ of $G$. The toughness $t(G)$ of $G$ is defined as $\min\left\{\frac{|S|}{c(G-S)}\right\}$, where…

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote $\mathscr{T}_n^{p, q}={T: T$ is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with…

Let $G$ be a graph with a spanning subgraph $F$, let $m$ be a positive integer, and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $$\Omega_m(G\setminus S)\le \sum_{v\in…

Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\leq n-g$. Moreover, we…

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai…

A signed graph $(G, \sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, \sigma)$. Let $c(G)$, $\alpha(G)$ and $r(G, \sigma)$ be the cyclomatic number, the independence number and the rank…

Bollob\'as and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$,…

Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every…

Let $TCG_n$ denote the coprime graph having vertex set $\{1,2,\ldots,n\}$ with any two vertices $i,j$ being adjacent if and only if $\gcd(i,j)=1$. In this article, we first study some structural properties of $TCG_n$. We study the vertex…

We investigate the parameterized complexity of the recognition problem for the proper $H$-graphs. The $H$-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph $H$, and the properness means that the…

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce…

For an $n \times n$ matrix $A$, let $q(A)$ be the number of distinct eigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let $\mathcal{S}(G)$ be the set of all real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for…

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on…

In 2019, Dvo\v{r}\'{a}k asked whether every connected graph $G$ has a tree decomposition $(T, \mathcal{B})$ so that $T$ is a subgraph of $G$ and the width of $(T, \mathcal{B})$ is bounded by a function of the treewidth of $G$. We prove that…

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by only keeping the largest distances for each row and each column, whereas the remaining entries become zero. This matrix is also called the…

Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $\sigma$ of $G$ is the signed graph $G^{\sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)\setminus X$. Let $A(G^{\sigma})$ be the adjacency…