Spectral Diameter Estimates for k-Regular Graphs

ON GENERALIZED k-DIAMETER OF k-REGULAR k-CONNECTED GRAPHS

In this paper, motivated by the study of the wide diameter and the Rabin number of graphs, we define the generalized k-diameter of k-connected graphs, and show that every k-regular k-connected graph on n vertices has the generalized k-diameter at most n/2 and this upper bound cannot be improved when n = 4k − 6 + i(2k − 4).

Regular vertex diameter critical graphs

A graph is called vertex diameter critical if its diameter increases when any vertex is removed. Regular vertex diameter critical graphs of every valency k ≥ 2 and diameter d ≥ 2 exist, raising the question of identifying the smallest such graphs. We describe an infinite family of k-regular vertex diameter critical graphs of diameter d with at most kd+ (2k − 3) vertices. This improves the previ...

Improving diameter bounds for distance-regular graphs

In this paper we study the sequence (ci)0≤i≤d for a distance-regular graph. In particular we show that if d ≥ 2j and cj = c > 1 then c2j > c holds. Using this we give improvements on diameter bounds by Hiraki and Koolen [5], and Pyber [8], respectively, by applying this inequality.

Spectral and Geometric Properties of k-Walk-Regular Graphs

Let us consider a connected graph G with diameter D. For a given integer k between 0 and D, we say that G is k-walk-regular if the number of walks of length between vertices u, v only depends on the distance between u and v, provided that such a distance does not exceed k. Thus, in particular, a 0-walk-regular graph is the same as a walk-regular graph, where the number of cycles of length roote...

Tight Estimates for Eigenvalues of Regular Graphs