Related papers: A classification of $Q$-polynomial distance-regula…

The gonality of a graph measures how difficult it is to move chips around the entirety of a graph according to certain chip-firing rules without introducing debt. In this paper we study the gonality of circulant graphs, a class of…

In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2/k<=C. This extends a conjecture of Bannai and Ito.

The line graph $\Gamma$ of a multi-graph $\Delta$ is the graph whose vertices are the edges of $\Delta$, where two such edges are adjacent if and only if they meet in a single vertex of $\Delta$. We provide several characterizations of such…

For a graph $\Gamma$, a positive integer $s$ and a subgroup $G\leq \Aut(\Gamma)$, we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of…

Let $G$ be a finite, connected graph. The eccentricity of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the arithmetic mean of the eccentricities of the vertices of $G$. We…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix…

A graph $\Gamma$ is said to be distance-balanced if for any edge $uv$ of $\Gamma$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$, and it is called nicely distance-balanced if…

In a graph $G$ of maximum degree 3, let $\gamma(G)$ denote the largest fraction of edges that can be 3 edge-coloured. Rizzi \cite{Riz09} showed that $\gamma(G) \geq 1-\frac{2\strut}{\strut 3 g_{odd}(G)}$ where $g_{odd}(G)$ is the odd girth…

As it is well known, the spectrum $ {\rm sp\,} \Gamma$ (of the adjacency matrix $A$) of a graph $\Gamma$, with $d$ distinct eigenvalues other than its spectral radius $\lambda_0$, usually provides a lot of information about the structure of…

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on $n$ vertices of diameter greater than two is at least $n/C$ for some universal constant…

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…

The alliance polynomial of a graph $\Gamma$ with order $n$ and maximum degree $\delta_1$ is the polynomial $A(\Gamma; x) = \sum_{k=-\delta_1}^{\delta_1} A_{k}(\Gamma) \, x^{n+k}$, where $A_{k}(\Gamma)$ is the number of exact defensive…

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H^2. Determining if a given graph G is the square of some graph is not easy…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

Let the Kneser graph $K$ of a distance-regular graph $\Gamma$ be the graph on the same vertex set as $\Gamma$, where two vertices are adjacent when they have maximal distance in $\Gamma$. We study the situation where the Bose-Mesner algebra…

The $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $\gamma(G)$) intersect in a…

Let $\Gamma$ denote a finite, connected graph with vertex set $X$. Fix $x \in X$ and let $\varepsilon \ge 3$ denote the eccentricity of $x$. For mutually distinct scalars $\{\theta^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix…

Given a graph $\Gamma$, its auxiliary \emph{square-graph} $\square(\Gamma)$ is the graph whose vertices are the non-edges of $\Gamma$ and whose edges are the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $\Gamma$. We…

Let $\Gamma=(V,E)$ be a graph of order $n$. A {\em closed distance magic labeling} of $\Gamma$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x) = r$ for all…