Related papers: Geometric and spectral analysis on weighted digrap…

This paper studies generalizations of the concept of acyclic orientations to arc-weighted orientations. These lead to four types of variations of strict degeneracy of graphs. Some of these variations are studied in the literature under…

Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…

A new class of simple symmetric digraphs called $\mathcal{D}$ is defined and studied here. Any digraph in $\mathcal{D}$ has the property that each non-pendant vertex is adjacent to at least one pendant vertex. A graph theoretical…

Metric networks are network-shaped, one-dimensional structures on which one can solve differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide…

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised…

In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of $n \times n$ involutions with spectrum equal to $\{ \lambda_1^{n-2}, \lambda_2^{2} \}$ for some $\lambda_1$ and some…

Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of cooperative control problems, including formation control, distributed averaging, and optimization. We refer to a digraph…

We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$…

It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\mathbf{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then…

Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different…

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…

The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any…

Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the…

Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ with the property that $f(d, (1+o(1))np, (1+o(1))np)=(1+o(1))f(d, np, np)$ for $d=1,2$. Let $G$ be a graph, $d_i$ denote the degree of a vertex $i$ of $G$ and $D(i, j)$…

Let $D(G)$ and $D^Q(G)= Diag(Tr) + D(G)$ be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph $G$, respectively, where $Diag(Tr)=\textrm{diag}(D_1,D_2,$ $\ldots,D_n)$ be the diagonal matrix…

Analyzing the spectral behavior of random matrices with dependency among entries is a challenging problem. The adjacency matrix of the random $d$-regular graph is a prominent example that has attracted immense interest. A crucial spectral…

The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices X and Y such that there are no arcs from X and Y . For a subclass of eulerian digraphs, we give…

Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold…

Let $D$ be an digraph of order $n$ with adjacency matrix $A(D)$ and outdegree matrix $\Delta^+=\Delta^+(D)$. Then the Laplacian and signless Laplacian matrices of $D$ are respectively defined as $L(D)=\Delta^+-A(D)$ and…

Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The…