Related papers: The rank of a signed graph in terms of girth

Suppose that $\Gamma=(G, \sigma)$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $\Gamma$ are called positive inertia index, negative inertia index and…

Let $\Gamma$ be a signed graph. The number of negative eigenvalues of the adjacency matrix of $\Gamma$ is called the negative inertia index of $\Gamma$, which is denoted by $i_-(\Gamma)$. The length of the shortest cycle contained in…

Let $G^\sigma=(G,\sigma)$ be a connected signed graph and $A(G^\sigma)$ be its adjacency matrix. The positive inertia index of $G^\sigma$, denoted by $p^{+}(G^\sigma)$, is defined as the number of positive eigenvalues of $A(G^\sigma)$.…

Let $G^{\sigma}$ be an oriented graph and $S(G^{\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\sigma}$. The skew-rank of $G^{\sigma}$, denoted by $sr(G^{\sigma})$, is the rank of $S(G^{\sigma})$.…

Let $G^{\sigma}=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $\eta(G,\sigma)$ be the nullity of $(G,\sigma)$. He et al. [Bounds for the matching number and…

A weighted graph $G^{\omega}$ consists of a simple graph $G$ with a weight $\omega$, which is a mapping,$\omega$: $E(G)\rightarrow\mathbb{Z}\backslash\{0\}$. A signed graph is a graph whose edges are labeled with $-1$ or $1$. In this paper,…

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…

A signed graph $(G,\sigma)$ on $n$ vertices is called a \textit{parity signed graph} if there is a bijective mapping $f \colon V(G) \rightarrow \{1,\ldots,n\}$ such that $f(u)$ and $f(v)$ have same parity if $\sigma(uv)=1$, and opposite…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $\Sigma\subseteq E$. The edges in $\Sigma$ are called odd and the other edges of $E$…

We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the…

A signed graph $\Sigma = (G, \sigma)$ is a graph where the function $\sigma$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $\Sigma$, denoted by $A(\Sigma)$, is defined canonically. In a recent…

A signed graph is a graph $G$ associated with a mapping $\sigma: E(G)\to \{-1,+1\}$, denoted by $(G,\sigma)$. A $cycle$ of $(G,\sigma)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges,…

A $k$-regular graph of girth $g$ is called edge-girth-regular graph, shortly egr-graph, if each of its edges is contained in exactly $\lambda$ distinct $g-$cycles. An egr-graph is called extremal for the triple $(k, g, \lambda)$ if has the…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V=\{1,\ldots,n\}$ and $\Sigma\subseteq E$. The edges in $\Sigma$ are called odd edges and the other edges of $E$ even.…

The study on parity signed graphs was initiated by Acharya and Kureethara very recently and then followed by Zaslavsky etc.. Let $(G,\sigma)$ be a signed graph on $n$ vertices. If $(G,\sigma)$ is switch-equivalent to $(G,+)$ at a set of…

A signed graph $ (G, \Sigma)$ is a graph positive and negative ($\Sigma $ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, \Sigma)$ is to switch the signs of the edges incident to $v$. If one can obtain $…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,...,n\}$ and $\Sigma\subseteq E$. The edges in $\Sigma$ are called odd and the other edges even. By…

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected…

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend…

An edge-girth-regular graph $egr(v,k,g,\lambda)$, is a $k$-regular graph of order $v$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g$-cycles. An $egr(v,k,g,\lambda)$ is called extremal…