A lower bound is given for the harmonic mean of the growth in a finite undirected graph 1 in terms of the eigenvalues of the Laplacian of 1. For a connected graph, this bound is tight if and only if the graph is distance-regular. Bounds on the diameter of a ``sphere-regular'' graph follow. Finally,
On the relationship between the diameter and the size of a boundary of a directed graph
โ Scribed by Shuji Jimbo; Akira Marouka
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 480 KB
- Volume
- 50
- Category
- Article
- ISSN
- 0020-0190
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
## Abstract We generalize the concept of the diameter of a graph __G__ = (__N, A__) to allow for location of points not on the nodes. It is shown that there exists a finite set of candidate points which determine this __generalized diameter.__ Given the matrix of shortest paths, an __o__ (|__A__|^2
## We prove the following theorem. If G b a connected finite graph of order p, and S is a k-subset of V(G) (where k 2 2), then there is a pair of vertices in S which are at a dbtance ~2 [(p -1)/k] if k does not divide p, and ~2 I@ -1)/k j + 1 otherwise.
We provide upper estimates on the spectral radius of a directed graph. In particular w e prove that the spectral radius is bounded by the maximum of the geometric mean of in-degree and out-degree taken over all vertices.
## Abstract In this note, we show how the determinant of the distance matrix __D(G__) of a weighted, directed graph __G__ can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks __G~i~__ of __G__. In particular, when cof __D(G__), the sum of the cofactors of _