Related papers: The Laplacian spectral excess theorem for distance…

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph is distance-biregular when it is…

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…

The Spectral Excess Theorem (SPET) for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. Recently, some local or global approaches to the SPET…

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the…

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be…

It is known that every distance-regular digraph is connected and normal. An interesting question is: when is a given connected normal digraph distance-regular? Motivated by this question first we give some characterizations of weakly…

The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal…

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of…

Let $\Gamma$ be a distance-regular graph with diameter $d$ and Kneser graph $K=\Gamma_d$, the distance-$d$ graph of $\Gamma$. We say that $\Gamma$ is partially antipodal when $K$ has fewer distinct eigenvalues than $\Gamma$. In particular,…

The normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}(G)$ of a graph $G$ is a natural generalization of the normalized Laplacian matrix, arising from the matrix of pairwise distances between vertices rather than the adjacency…

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$…

We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families…

Let $G$ be a simple connected undirected graph. The Laplacian spectral ratio of $G$, denoted by $R_L(G)$, is defined as the quotient between the largest and second smallest Laplacian eigenvalues of $G$, which is closely related to the…

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be…

Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity,…

For a connected graph $G$, the distance Laplacian spectral radius of $G$ is the spectral radius of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is a diagonal matrix of vertex…

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i(e)=|d(u)-d(v)|$, where $d(\cdot)$ is the vertex degree. The irregularity $I(G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was…

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the…

Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of…

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)$ and $D^{Q}(G)$, respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of…