Extended Schatten norms of random graphs and Nikiforov conjecture

Abstract

In this paper we give mean of $p$-th degree of singular values and upper bound of geometric mean for almost all graphs. We prove three theorems about a conjecture of V. Nikiforov for Schatten $p$-norm of graphs when $p>2$. We prove that the conjecture is true when $p$ is an even integer or when graph is a tree or a strongly regular graph with certain parameters. The strongly regular graphs with these parameters are graphs with maximal energy.

Keywords

• singular values of graph
• Schatten norms
• random graph
• energy of graph

References

1. [1] L. Arnold, On the asymptotic distribution of the eigenvalues of random matrices, J. Math. Anal. Appl. 20, 262-268, 1967.
2. [2] T. Chadjipantelis, S. Kounias and C. Moyssiadis, The maximum determinant of $21\times21$ $(+ 1,-1)$-matrices and D-optimal designs, J. Statist. Plan. Inference, 16, 167-178, 1987.
3. [3] U. Grenander, Probability on algebraic structures, Wiley, New York, 1963.
4. [4] J.H. Koolen and V. Moulton, Maximal energy graphs, Adv. Appl. Math. 26, 47-52, 2001.
5. [5] N. Nguyen and A.J. Miller, A review of some exchange algorithms for constructing discrete D-optimal designs, Comput. Stat. Data Anal. 14, 489-498, 1992.
6. [6] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326, 1472- 1475, 2007.
7. [7] V. Nikiforov, Extremal norms of graphs and matrices, J. Math. Sci. 182, 164-174, 2012.
8. [8] V. Nikiforov, Beyond graph energy: Norms of graphs and matrices, Linear Algebra Appl. 506, 82-138, 2016.

9. [9] R. Sitter and B. Torsney, Optimal designs for binary response experiments with two design variables, Statist. Sinica, 5, 405-419, 1995.
10. [10] E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. (2) 67, 325-327, 1958.