Related papers: The subgraph eigenvector centrality of graphs

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with…

The $k$-token graph $F_k(G)$ of a graph $G$ on $n$ vertices is the graph whose vertices are the ${n\choose k}$ $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices…

If $G$ is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs $G_{1}$ and $G_{2}$ is a graph $G=G_{1}\odot G_{2}$ with $V(G)=V(G_{1})\cup…

Given a graph and one of its weighted Laplacian matrix, a Fiedler vector is an eigenvector with respect to the second smallest eigenvalue. The Fiedler vectors have been used widely for graph partitioning, graph drawing, spectral clustering,…

In an era where accumulating data is easy and storing it inexpensive, feature selection plays a central role in helping to reduce the high-dimensionality of huge amounts of otherwise meaningless data. In this paper, we propose a graph-based…

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph $G$ is a median graph if, for all $\mu, u,v\in…

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…

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…

An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq2$ is a trivial necessary condition for a graph to have an even factor, where $\delta(G)$ is the minimum…

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of…

Identifying the most influential nodes in networked systems is of vital importance to optimize their function and control. Several scalar metrics have been proposed to that effect, but the recent shift in focus towards network structures…

Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $\beta$-subgraph…

We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network…

We show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set, the most extreme example being networks with a cut vertex. Three distinct types of…

We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$,…

Given a graph $G=(V,E)$ on $n$ vertices and an integer $k$ between 1 and $n-1$, the $k$-token graph $F_k(G)$ has vertices representing the $k$-subsets of $V$, and two vertices are adjacent if their symmetric difference is the two…

The rank $r(G)$ of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the nullity $\eta(G)$ of $G$ is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this paper, we prove that if $G$ is a connected graph of order $n$ with…

Let G be a finite simple graph with automorphism group A(G). Then a spanning subgraph U of G is a fixing subgraph of G if G contains exactly $| A(G)|/ | A(G) \cap A(U)| $ subgraphs isomorphic to U: the graph G must always contain at least…

The edge betweenness centrality of an edge is loosely defined as the fraction of shortest paths between all pairs of vertices passing through that edge. In this paper, we investigate graphs where the edge betweenness centrality of edges is…