From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs
✍ Scribed by M.A Fiol; E Carriga
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 488 KB
- Volume
- 71
- Category
- Article
- ISSN
- 0095-8956
No coin nor oath required. For personal study only.
✦ Synopsis
The local adjacency polynomials can be thought of as a generalization, for all graphs, of (the sums of ) the distance polynomials of distance-regular graphs. The term local'' here means that we see'' the graph from a given vertex, and it is the price we must pay for speaking of a kind of distance-regularity when the graph is not regular. It is shown that when the value at * (the maximum eigenvalue of the graph) of the local adjacency polynomials is large enough, then the eccentricity of the base vertex tends to be small. Moreover, when such a vertex is ``tight'' (that is, the value of a certain polynomial just fails to satisfy the condition) and fulfils certain additional extremality conditions, then all the polynomials attain their maximum possible values at *, and the graph turns out to be pseudo-distance-regular around the vertex. As a consequence of the above results, some new characterizations of distance-regular graphs are derived. For example, it is shown that a regular graph 1 with d+1 distinct eigenvalues is distance-regular if, and only if, the number of vertices at distance d from any given vertex is the value at * of the highest degree member of an orthogonal system of polynomials, which depend only on the spectrum of the graph. 1997 Academic Press 1. INTRODUCTION Recently, much work has been done on bounding some distance-related parameters of a graph, such as the eccentricity of a vertex, the diameter, and the radius, in terms of (part of ) the spectrum. See, for instance, the works of Alon and Milman [1], Mohar [24], Chung [4], Sarnak [27], Chung et al. [5], Quenell [25], Van Dam and Haemers [9], Delorme and Sole [12], Delorme and Tillich [13], Kahale [23], and Yebra and the present authors [15, 14]. In some of these papers, different families of polynomials, such as the Chebyshev polynomials and their discrete version, article no. TB971778 162 0095-8956Â97 25.00