A new conjecture in terms of Laplacian eigenvalues and the degree sequence of a graph

Abstract

Let $d_1\geq d_2\geq\cdots\geq d_n$ be the degree sequence of a graph $G$ of order $n$ and $\mu_1\geq\mu_2\geq\cdots\geq\mu_n=0$ be the Laplacian eigenvalues of $G$. In this paper, we propose a new conjecture that for any graph $G$ except for $C_{4k+1}(k\in Z^+)$, $$\sum_{\mu_i\geq 2}(\mu_i-2)^2\leq(1-\frac{1}{d_1})\sum_{i=1}^{n}d_i(d_i-1).$$
We also prove this conjecture is true for the star, the path, the strongly regular graph, the threshold graph, the barbell graph and the complete bipartite graph, respectively.