Related papers: The least eigenvalue of graphs whose complements a…
The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy…
For a simple graph, the minimum rank problem is to determine the smallest rank among the symmetric matrices whose off-diagonal nonzero entries occur in positions corresponding to the edges of the graph. Bounds on this minimum rank (and on…
A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by…
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 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional…
The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$…
For a graph $G$ with vertex set $V$, let N($G$) denote the number of nonempty subsets of $V$ that induce a connected graph in $G$. In this paper, we focus on determining N($G$) for $G$ in the family $\mathbb{B}_n$ of $n$-vertex bicyclic…
Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called $c$-closure, for a parameterized study of clique enumeration problems. A graph $G$ is $c$-closed if every pair of vertices with at least $c$ common neighbors is adjacent.…
A labelled, undirected graph is a graph whose edges have assigned labels, from a specific set. Given a labelled, undirected graph, the well-known minimum labelling spanning tree problem is aimed at finding the spanning tree of the graph…
Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P = P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue…
A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $\Gamma$ is the minimum…
Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest…
The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power…
In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds…
The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$.…
An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic…
For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…
The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…
Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank…