Conjectured bounds for the sum of squares of positive eigenvalues of a graph

Clive Elphick, Miriam Farber, Felix Goldberg, Pawel Wocjan

Abstract: A well known upper bound for the spectral radius of a graph, due to Hong, is that μ12≤2m−n+1<math><msubsup is="true"><mrow is="true"><mi is="true">μ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></msubsup><mo is="true">≤</mo><mn is="true">2</mn><mi is="true">m</mi><mo is="true">−</mo><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn></math> if δ≥1<math><mi is="true">δ</mi><mo is="true">≥</mo><mn is="true">1</mn></math>. It is conjectured that for connected graphs n−1≤s+≤2m−n+1<math><mi is="true">n</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">≤</mo><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">+</mo></mrow></msup><mo is="true">≤</mo><mn is="true">2</mn><mi is="true">m</mi><mo is="true">−</mo><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn></math>, where s+<math><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">+</mo></mrow></msup></math> denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q<math><mi is="true">q</mi></math>-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.