arXiv:2007.01300v2 [math.CO] 11 Jul 2020 INTEGRAL EQUIENERGETIC NON-ISOSPECTRAL CAYLEY GRAPHS RICARDO A. PODESTÁ, DENIS E. VIDELA July 14, 2020 Abstract. We prove that the Cayley graphs X (G, S ) and X + (G, S ) are equienergetic for any abelian group G and any symmetric subset S. We focus on two subfamilies: uni- tary Cayley graphs G R = X (R, R * ), where R is a commutative ring, and semiprimitive generalized Paley graphs Γ(k,q)= X (F q , {x k : x k ∈ F * q }). We prove that under mild conditions, {G R ,G + R } and {Γ(k,q), Γ + (k,q)} are pairs of equienergetic non-isospectral graphs (generically integral, connected and non-bipartite). Then, we obtain conditions such that {G R , ¯ G R } and {Γ(k,q), ¯ Γ(k,q)} are equienergetic non-isospectral graphs. Fi- nally, we characterize all (integral) equienergetic non-isospectral triples {G R ,G + R , ¯ G R } and {Γ(k,q), Γ + (k,q), ¯ Γ(k,q)} such that all the graphs are also Ramanujan. 1. Introduction This paper deals with the spectrum and the energy of Cayley graphs and Cayley sum graphs. We mainly focus on two subfamilies: unitary Cayley (sum) graphs over rings and generalized Paley (sum) graphs. Our main goal is to give a general construction of infinite pairs of integral equienergetic non-isospectral graphs with some nice extra properties like being connected, non-bipartite or Ramanujan (or all of them). One of the graphs of the pairs can be taken either with or without loops. If Γ is a graph of n vertices, the eigenvalues of Γ are the eigenvalues {λ i } n i=1 of its adjacency matrix. The spectrum of Γ, denoted Spec(Γ) = {[λ i 1 ] e 1 ,..., [λ is ] e is }, is the set of all the different eigenvalues {λ i j } of Γ counted with their multiplicities {e i j }. The spectrum is symmetric if for every eigenvalue λ, its opposite −λ is also an eigenvalue with the same multiplicity as λ. The graph is called integral if Spec(Γ) ⊂ Z, i.e. if all of its eigenvalues are integers. The energy of Γ is defined by E(Γ) = ∑ n i=1 |λ i |. We refer to the books [5] or [6] for a complete viewpoint of spectral theory of graphs, and to [8] for a survey on energy of graphs. Equienergetic non-isospectral graphs. Let Γ 1 and Γ 2 be two graphs with the same number of vertices. The graphs are called isospectral (or cospectral ) if Spec(Γ 1 )= Spec(Γ 2 ) and equienergetic if E(Γ 1 )= E(Γ 2 ). It is clear by the definitions that isospectrality implies equienergeticity, but the converse is false. Thus, we are interested in the construction of Key words and phrases. Equienergetic, non-isospectral, unitary Cayley graphs, generalized Paley graphs, Ramanujan. 2010 Mathematics Subject Classification. Primary 05C25; Secondary 05C50, 05C75. Partially supported by CONICET and SECyT-UNC. 1