Graph with the smallest diameter.

Question

Consider a graph with $N$ vertexes where each vertex has at most $k$ edges. I assume that $k < N$. What is the graph which have above property and has the smallest diameter?

Also, could you suggest good books in graph theory. Thanks.

You would need to ask your second question separately. But you shouldn't, because it would be a duplicate: http://math.stackexchange.com/q/27480/12952 — Alexander Gruber, Feb 28 '13 at 20:47

Boris Novikov · Answer 1 · 2013-06-12T07:31:04.517

2

Douglas B. West, Introduction to Graph Theory (2nd edition), Ex.2.1.60:

Let $G$ be a graph with diameter $d$ and maximum degree $k$. Prove that $n(G) \le 1 + [(k - 1)^d - 1]\frac{k}{k - 2}$. (Comment: Equality holds for the Petersen graph.)

Addendum: See also the comment of Andrew Salmon below.

edited Jun 12 '13 at 07:31

answered Feb 28 '13 at 20:47

Boris Novikov

17,754

To add to this list: Diestel's Graph Theory is quite good as well (with different emphasis--less emphasis on algorithms than West but more on the graph minor theorem and topics outside coloring, matching, and connectivity), and the author has generously provided a version of the book online. – A.S Jun 12 '13 at 06:43
@Andrew Salmon: Yes, thank you. – Boris Novikov Jun 12 '13 at 07:31

score 0 · Answer 2 · answered Jul 02 '14 at 16:16

This question is quite difficult. The upper bound on the number of vertices given above by @Boris Novikov coincides with Moore's bound in the case of the Petersen graph. Moore's bound is not only achieved by the Petersen graph, but also by the Hoffman-Singleton graph, and in general, by the so-called Moore graphs. Unfortunately, there are very few Moore graphs. In most other cases, i.e. when Moore's bound cannot be reached, the optimal graph is not known. For more details see "Moore Graphs and Beyond: A survey of the Degree/Diameter Problem", by Mirka Miller and Josef Siran (Elec. J. Combinatorics, Dynamic Survey 14).

Graph with the smallest diameter.

2 Answers2