A new infinite series of regular uniformly geodetic code graphs

Abstract

We construct vertex-transitive graphs Γ, regular of valency k= n 2+ n+1 on v=2( 2n n ) vertices, with integral spectrum, possessing a distinguished complete matching such that contracting the edges of this matching yields the Johnson graph J(2 n, n) (of valency n 2). These graphs are uniformly geodetic in the sense of Cook and Pryce (1983) ( F-geodetic in the sense of Ceccharini and Sappa (1986)), i.e., the number of geodesics between any two vertices only depends on their distance (and equals 4 when this distance is two). They are counterexamples to Theorem 3.15.1 of [1], and we show that there are no other counterexamples.