Related papers: Edge Connectivity, Packing Spanning Trees, and Eig…

Let $n$ be any positive integer, the friendship graph $F_n$ consist of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same…

Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,...,q\},$ $q \in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the…

For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal…

Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two distinct such cycles is the set of edges of $T$ that belong to both cycles. We consider…

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is…

Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…

Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $\langle x\rangle\subseteq \langle y\rangle$ or $\langle y\rangle\subseteq…

The problem of determining the smallest number of edges, $h(n;\bar{\kappa}\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\bar{\kappa}\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint…

An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of…

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the…

Let $G$ be a $d$-regular multigraph, and let $\lambda_2(G)$ be the second largest eigenvalue of $G$. In this paper, we prove that if $\lambda_2(G) < \frac{d-1+\sqrt{9d^2-10d+17}}4$, then $G$ is 2-edge-connected. Furthermore, for $t\ge2$ we…

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

A graph G=(V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers `d' and `D' such that each leaf `u' of T is a node of V and the edge `(u,v) belongs to E' iff `d <= d_T(u, v)…

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $\kappa'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components…

We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every…

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested…

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq…