Related papers: The signed graphs with symmetric spectra
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of…
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs.…
Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its adjacency matrix. For a non-complete bipartite graph $G$ with parts $X$ and $Y$, the {\em bipartite toughness} of $G$ is defined as…
A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles.…
Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a…
A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$…
A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is,…
We introduce the set $\mathcal{G}^{\rm SSP}$ of all simple graphs $G$ with the property that each symmetric matrix corresponding to a graph $G \in \mathcal{G}^{\rm SSP}$ has the strong spectral property. We find several families of graphs…
A factor of a graph is a spanning subgraph. Spectral sufficient conditions are provided via spectral radius and signless Laplacian spectral radius for graphs with (i) a matching of given size (particularly, $1$-factor) containing any given…
A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another…
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. Alazemi, Andeli\'c and Simi\'c conjectured that no chain graph shares a non-zero…
A mixed graph $G$ is a graph obtained from a simple undirected graph by orientating a subset of edges. $G$ is self-converse if it is isomorphic to the graph obtained from $G$ by reversing each directed edge. For two mixed graphs $G$ and $H$…
Let $\Gamma$ be a $G$-symmetric graph with vertex set $V$. We suppose that $V$ admits a $G$-partition $\mathcal{B} = \{ B_0, ... , B_b \}$, with parts of size $v$, and that the quotient graph induced on $\mathcal B$ is a complete graph of…
We completely determine the spectrum of an $I$-graph, that is, the eigenvalues of its adjacency matrix. We apply our result to prove known characterizations of connectedness and bipartiteness in $I$-graphs by using an spectral approach.…
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 graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph…
The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered…
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained…
Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse…
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of…