Related papers: On $s$-distance-transitive graphs

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least…

The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley…

We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the…

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to…

Let $\Gamma=\Gamma(A)$ denote a simple strongly connected digraph with vertex set $X$, diameter $D$, and let $\{A_0,A:=A_1,A_2,\ldots,A_D\}$ denote the set of distance-$i$ matrices of $\Gamma$. Let $\{R_i\}_{i=0}^D$ denote a partition of…

A graph $X$ is said to be {\it distance--balanced} if for any edge $uv$ of $X$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. A graph $X$ is said to be {\it strongly…

In this paper, we study self-complementary distance-regular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is self-complementary strongly regular. We also deal with…

In this paper we show that certain almost distance-regular graphs, the so-called $h$-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is called…

We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral…

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral…

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not…

In a recent paper, Chernikov and Starchenko prove that graphs defined in distal theories have strong regularity properties, generalizing previous results about graphs defined by semi-algebraic relations. We give a shorter, purely…

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…

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs…

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [8], the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph.…

We study transitivity properties of graphs with more than one end. We completely classify the distance-transitive such graphs and, for all $k \geq 3$, the $k$-CS-transitive such graphs.

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of $\G$ there are exactly $\gamma$ vertices of $\G$ which are…

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…