The extremal spectral radii of the arithmetical structures on paths

Abstract An arithmetical structure on a finite, connected graph G is a pair of vectors ( d , r ) with positive integer entries for which ( d i a g ( d ) − A ( G ) ) r T = 0 , where d i a g ( d ) = d i a g ( d 1 , d 2 , … , d n ) , A ( G ) is the adjacency matrix of G and the entries of r have no common factor. In this paper, we will study the spectral radii of arithmetical structures on the path P n and determine the arithmetical structures with the minimal and maximal spectral radius on P n .