Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex v such that d ( v , v ) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d ( u , v ) + d ( u , v ) = diam G for all u , v ∈ V ( G ). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3 d – 1 and conjectured that n ≥ 4 d – 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4 d – 4 there exists an even graph of order n and diameter d . Several ways of constructing new even graphs from known ones are presented.