Related papers: Connected signed graphs with given inertia indices…

For an integer $s\geq1$ and a graph $\Gamma$, a path $(u_0, u_1, \ldots, u_{s})$ of vertices of $\Gamma$ is called an {\em $s$-geodesic} if it is a shortest path from $u_0$ to $u_{s}$. We say that $\Gamma$ is {\em $s$-geodesic transitive}…

The equator of a graph is the length of a longest isometric cycle. We bound the order $n$ of a graph from below by its equator $q$, girth $g$ and minimum degree $\delta$ - and show that this bound is sharp when there exists a Moore graph…

Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be…

The principal ratio of a connected graph, denoted $\gamma(G)$, is the ratio of the maximum and minimum entries of its first eigenvector. Cioab\u{a} and Gregory conjectured that the graph on $n$ vertices maximizing $\gamma(G)$ is a kite…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

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…

Circular $r$-coloring of a signed graph $(G,\sigma)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a…

Let $\Gamma$ be an undirected and simple graph. A set $ S $ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma - S$ is disconnected and has at least two components containing cycles. If $\Gamma$ has a cyclic…

A transversal set of a graph $G$ is a set of vertices incident to all edges of $G$. The transversal number of $G$, denoted by $\tau(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is…

A graph $G$ is weakly $\gamma$-closed if every induced subgraph of $G$ contains one vertex $v$ such that for each non-neighbor $u$ of $v$ it holds that $|N(u)\cap N(v)|<\gamma$. The weak closure $\gamma(G)$ of a graph, recently introduced…

A signed graph is a graph whose edges are signed. In a vertex-signed graph the vertices are signed. The latter is called consistent if the product of signs in every circle is positive. The line graph of a signed graph is naturally…

A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed…

A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen…

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from…

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the…

A signed graph $\Gamma$ is said to be determined by its spectrum if every signed graph with the same spectrum as $\Gamma$ is switching isomorphic with $\Gamma$. Here it is proved that the path $P_n$, interpreted as a signed graph, is…

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

Let $G$ be a graph with adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Both $\lambda_1 + \lambda_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikv\'ari proved in 2022 that for odd girth 5…

A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be…