Invariant adjacency matrices of configuration graphs

Some graphs Γ have the following property P: the configuration graph (i.e. the non–collinearity graph) of the neighbourhood geometry of Γ is isomorphic to Γ. For instance, the ubiquitous Petersen graph satisfies P. The purpose of this paper is to reveal repercussions of property P on adjacency matrices A for Γ. This will be achieved in terms of invariance under (powers of) the following mapping Θ: denote by I and J the identity matrix and the all one matrix, respectively, both of order n = k2+1, and define Θ(A) := (k−1)I + J−A2. If k = 3 and A is an adjacency matrix for the Petersen graph, it is well known that Θ(A) = A. In 1960, Hoffman and Singleton showed that for arbitrary k the matrix equation Θ(A) = A has only very few solutions, namely for k = 2,3,7 and possibly for k = 57.We prove that the property P implies the existence of an integer m ⩾ 1 such that Θm(A) = A. We determine a standard form for such matrices which is invariant under the action of Θ. In particular, we characterize all solutions, on A, to Θm(A) = A for k = 3 and k = 4, where A is an adjacency matrix of some k–regular graph, solving a conjecture posed by the authors.