Related papers: On signed graphs whose spectral radius does not ex…

Let $\rho(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $\rho(G)>\sqrt{m}$, then…

A signed directed graph is a graph with sign and direction information on the edges. Even though signed directed graphs are more informative than unsigned or undirected graphs, they are more complicated to analyze and have received less…

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 \emph{signed graph} is a pair $\Gs$ in which $G$ is a finite simple graph and $\sigma:\E(G)\to\{+1,-1\}$ is a \emph{signature}. Following M\'a\v{c}ajov\'a--Raspaud- \v{S}koviera and Jin--Kang--Steffen, a \emph{proper coloring} of $\Gs$ is…

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least…

Let $G$ be a Cayley graph of the elementary abelian $2$-group $\mathbb{Z}_2^{n}$ with respect to a set $S$ of size $d$. We prove that for any such $G, S$ and $d$, the maximum degree of any induced subgraph of $G$ on any set of more than…

Let $\mathcal{G}_{n, \beta^*}$ $(\mathcal{G}^*_{n,\beta^*})$ be the set of all (connected) graphs of order $n$ with fractional matching number $\beta^*$. In this paper, the graphs with maximal spectral radius in $\mathcal{G}_{n,\beta^*}$…

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower…

A graph $G$ is $H$-free, if it contains no $H$ as a subgraph. A graph is said to be \emph{$H$-minor free}, if it does not contain $H$ as a minor. In recent years, Nikiforov asked that what is the maximum spectral radius of an $H$-free graph…

A perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a…

A graph $G$ is called {\em$F$-saturated} if $G$ does not contain $F$ as a subgraph but adding any missing edge to $G$ creates a copy of $F$. In this paper, we consider the spectral saturation problem for the linear forest $tP_4$, proving…

In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and…

Signed networks are graphs whose edges are labelled with either a positive or a negative sign, and can be used to capture nuances in interactions that are missed by their unsigned counterparts. The concept of balance in signed graph theory…

A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between…

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of…

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or…

A shortest circuit cover ${\cal F}$ of a bridgeless graph $G$ is a family of circuits that covers every edge of $G$ and is of minimum total length. The total length of a shortest circuit cover ${\cal F}$ of $G$ is denoted by $SCC(G)$. For…

In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $\lambda\leq -1$, we show that the…

The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are often tighter and are applicable to a larger class of digraphs than previously reported…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…