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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…

Given a graph $G$, let $\lambda_3$ denote the third largest eigenvalue of its adjacency matrix. In this paper, we prove various results towards the conjecture that $\lambda_3(G) \le \frac{|V(G)|}{3}$, motivated by a question of Nikiforov.…

We investigate the question how `small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of…

Eigenvalues of a graph are the eigenvalues of the corresponding (0,1)-adjacency matrix. The second largest eigenvalue lambda_2 provides significant information on characteristics and structure of graphs. Therefore, finding bounds for…

Every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$. The result is sharp.

The $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $\gamma(G)$) intersect in a…

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to…

We prove that, if $\Gamma$ is a finite connected $3$-valent vertex-transitive, or $4$-valent vertex- and edge-transitive graph, then either $\Gamma$ is part of a well-understood family of graphs, or every non-identity automorphism of…

The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of…

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family…

Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G)…

Let $G$ be a graph of order $n$. For every $v\in V(G)$, let $E_G(v)$ denote the set of all edges incident with $v$. A signed $k$-submatching of $G$ is a function $f:E(G)\longrightarrow \{-1,1\}$, satisfying $f(E_G(v))\leq 1$ for at least…

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is a function $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each edge $e\in E$, where $N[e]$ is the set of edges in $G$ that share at…

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as…

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so…

A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed…

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$,…

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…

In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies.