Related papers: Further results on the nullity of signed graphs
Signed graphs have their edges labeled either as positive or negative. $\rho(M)$ denote the $M$-spectral radius of $\Sigma$, where $M=M(\Sigma)$ is a real symmetric graph matrix of $\Sigma$. Obviously,…
Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with eigenvalues $\mu_1,\mu_2,\ldots,\mu_n.$ We define the Estrada index of a signed graph $\Gamma$ as $EE(\Gamma)=\sum_{i=1}^ne^{\mu_i}$. We characterize the signed unicyclic graphs…
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial $c_\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\Gamma$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an…
The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values…
For a graph $G = (V, E)$, the $\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent if they differ by a single vertex and the two…
We describe parity labelings of signed graphs; equivalently, cuts of the underlying graph that have nearly equal sides. We characterize the balanced signed graphs which are parity signed graphs. We give structural characterizations of all…
A signed graph $(G,\Sigma)$ is a graph $G$ together with a set $\Sigma \subseteq E(G)$ of negative edges. A circuit is positive if the product of the signs of its edges is positive. A signed graph $(G,\Sigma)$ is balanced if all its…
For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of…
A signed graph $(G, \Sigma)$ is a graph $G$ and a subset $\Sigma$ of its edges which corresponds to an assignment of signs to the edges: edges in $\Sigma$ are negative while edges not in $\Sigma$ are positive. A closed walk of a signed…
We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric…
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…
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…
In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…
The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination…
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the…
Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically…
The graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs $\mathcal{L}^k(G)$ with all equal negative eigenvalues $-2$ are characterized for $k\ge…
The bipartition polynomial of a graph is a generalization of many other graph polynomials, including the domination, Ising, matching, independence, cut, and Euler polynomial. We show in this paper that it is also a powerful tool for proving…