On regular graphs equienergetic with their complements

RICARDO A. PODESTÁ; DENIS E. VIDELA

LINEAR AND MULTILINEAR ALGEBRA

TAYLOR & FRANCIS LTD

Lugar: Londres; Año: 2023 vol. 71 p. 422 - 456

0308-1087

We give necessary and sufficient conditions on the parameters of a regular graph Γ (with or without loops) such that E(Γ)=E(Γ¯¯¯). We study complementary equienergetic cubic graphs obtaining classifications up to isomorphisms for connected cubic graphs with single loops (5 non-isospectral pairs) and connected integral cubic graphs without loops (Γ=K3□K2 or Q3). Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs Cr(n) or C4. Next, for the family of strongly regular graphs Γ we characterize all possible parameters srg(n,k,e,d) such that E(Γ)=E(Γ¯¯¯). Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e. has OA parameters). We also characterize all complementary equienergetic pairs of graphs of type C(2), C(3) and C(5) in Cameron´s hierarchy (the cases C(1) in the non-bipartite case and C(4) are still open). Finally, we consider unitary Cayley graphs over rings GR=X(R,R∗). We show that if R is a finite Artinian ring with an even number of local factors, then GR is complementary equienergetic if and only if R=Fq×Fq´ is the product of 2 finite fields.