Nonexistence of Some 4-regular Integral Graphs

A graph is called integral if all its eigenvalues are integers. The quest for integral graphs was initiated by F. Harary and A. J. Schwenk [9]. All such cubic graphs were obtained by D. Cvetković and F. C. Bussemaker [2, 1], and independently by A. J. Schwenk [10]. There are exactly thirteen cubic integral graphs. In fact, D. Cvetković [2] proved that the set of regular integral graphs of a fixed degree is finite. Z. Radosavljević and S. Simić [11] determined all 13 nonregular nonbipartite integral graphs whose maximum degree equals 4. Recently, 4-regular integral graphs attracted some attention. D. Stevanović [12] determined all 24 4-regular integral graphs avoiding ±3 in the spectrum. All known 4-regular integral graphs are collected in paper [8] by D. Cvetković, S. Simić and D. Stevanović. The potential spectra of bipartite 4-regular integral graphs are also determined in [8]. They are quite numerous and it cannot be expected that all 4-regular integral graphs will be determined in near future. In this note we obtain nonexistence results for some of these potential spectra. It follows from these results that, except for 5 exceptional spectra, bipartite 4regular integral graph has at most 1260 vertices. As a corollary, nonbipartite 4regular integral graph G has at most 630 vertices, unless G×K2 has one of these exceptional spectra.