This paper presents bounds for the variation of the spectral radius (G) of a graph G after some perturbations or local vertex/edge modifications of G. The perturbations considered here are the connection of a new vertex ...

The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph Γ with d+1 distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at distance d) of each ...

Given a simple connected graph $\Gamma$ and a subset of its vertices $C$, the pseudo-distance-regularity around $C$ generalizes, for not necessarily regular graphs, the notion of completely regular code. Up to now, most ...

Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first propose two dual concepts ...

Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a ...

Let G be a connected graph with vertex set V and a weight function that assigns a nonnegative number to each of its vertices. Then, the -moment of G at vertex u is de ned to be M G(u) = P v2V (v) dist(u; v), where ...

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular ...

Considering a connected graph $G$ with diameter $D$, we say that it is \emph{$k$-walk-regular}, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between vertices $u$ and $v$ only depends ...