A new infinite series of regular uniformly geodetic code graphs

We construct vertex-transitive graphs Γ, regular of valency k=n2+n+1 on v=2(2nn) vertices, with integral spectrum, possessing a distinguished complete matching such that contracting the edges of this matching yields the Johnson graph J(2n, n) (of valency n2). 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