Related papers: The matching energy of random graphs
Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1 \geq \cdots \geq\lambda_n$. Let \[s^+(G)=\sum_{\lambda_i>0} \lambda_i^2, \qquad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called…
The network alignment problem asks for the best correspondence between two given graphs, so that the largest possible number of edges are matched. This problem appears in many scientific problems (like the study of protein-protein…
Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…
The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we present a new method to compare the energies of two $k$-subdivision bipartite graphs on some cut edges. As the…
In this paper, we determine the Randic' energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs…
In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects,…
Let $\Gamma$ be a graph with the adjacency matrix $A$. The energy of $\Gamma$ is the sum of the absolute values of the eigenvalues of $A$. In this article we compute the energies of the commuting graphs of some finite groups and discuss…
We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson…
Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…
In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and…
The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let…
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its…
The energy $E(G)$ of a simple graph $G$ is the sum of absolute values of the eigenvalues of its adjacency matrix. A borderenergetic graph of order $n \in \mathbb{N}$ is any noncomplete graph~$G$ such that $E(G) = E(K_n) = 2n - 2$. Here we…
In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still…
In 2018, Arizmendi and Juarez introduced the concept of energy of a vertex, a novel approach allowing the total energy of a graph to be expressed as the sum of the energies of its individual vertices. In this article, we extend the notion…
For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$.…
The energy of a simple graph $G$ arising in chemical physics, denoted by $\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice…
We solve the conjecture posed by Fill, Scheinerman and Singer-Cohen and show the equivalence of the sharp threshold functions of the random intersection graph G(n,m,p) with $m >= n^3$ and a graph in which each edge appears independently.…
We establish the existence of free energy limits for several combinatorial models on Erd\"{o}s-R\'{e}nyi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. For a variety of models, including…
In the \emph{graph matching} problem we observe two graphs $G,H$ and the goal is to find an assignment (or matching) between their vertices such that some measure of edge agreement is maximized. We assume in this work that the observed pair…