Andrásfai graph

In graph theory, an Andrásfai graph is a triangle-free, circulant graph named after Béla Andrásfai.

Andrásfai graph
Named after	Béla Andrásfai
Vertices	${\displaystyle 3n-1}$
Edges	${\displaystyle {\frac {n(3n-1)}{2}}}$
Diameter	2
Properties	Triangle-free Circulant
Notation	And(n)
Table of graphs and parameters

Two drawings of the And(4) graph

Properties

The Andrásfai graph And(n) for any natural number n ≥ 1 is a circulant graph on 3n − 1 vertices, in which vertex k is connected by an edge to vertices k ± j, for every j that is congruent to 1 mod 3. For instance, the Wagner graph is an Andrásfai graph, the graph And(3).

The graph family is triangle-free, and And(n) has an independence number of n. From this the formula R(3,n) ≥ 3(n − 1) results, where R(n,k) is the Ramsey number. The equality holds for n = 3 and n = 4 only.

The Andrásfai graphs were later generalized.^[1][2]