Spectral Characterization of Some Generalized Odd Graphs

Suppose G is a connected, k-regular graph such that Spec…G† ˆ Spec…G† where G is a distance-regular graph of diameter d with parameters a1 ˆ a2 ˆ ˆ adÿ1 ˆ 0 and ad > 0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of G in terms of the notion of Ho ̈man Polynomials. Furthermore, G is isomorphic to G if G is one of the odd polygon C2d‡1, the Odd graph Od‡1, the folded …2d ‡ 1†-cube, the coset graph of binary Golay code …d ˆ 3†, the Ho ̈manSingleton graph …d ˆ 2†, the Gewirtz graph …d ˆ 2†, the Higman-Sims graph …d ˆ 2†, or the second subconstituent of the Higman-Sims graph …d ˆ 2†.