Signless Laplacian power-sum conjecture for graphs with bounded vertex connectivity

Let $\mathcal{F}_n$ be the family of graphs on $n$ vertices, let $\mathcal{V}_n^k$ denote the graphs in $\mathcal{F}_n$ with vertex connectivity at most $k$, and let $n,k$ be positive integers with $1\leq k\leq n-1$. For a graph $G$, write $Q(G)$ for its signless Laplacian matrix, let $q_1,\dots,q_n$ be the eigenvalues of $Q(G)$, and define

$$S_\alpha(G)=\sum_{i=1}^n q_i^\alpha.$$

Let $b_\alpha(n,k)$ denote the bound defined earlier in the paper. Signless Laplacian power-sum conjecture in $\mathcal{V}_n^k$. For $G\in\mathcal{V}_n^k$ and $\alpha<1$:

1. If $0<\alpha<1$, then

$$S_\alpha(G)\leq b_\alpha(n,k),$$

with equality if and only if $G=K_k\vee(K_1\cup K_{n-k-1})$.

2. If $\alpha<0$, then

$$S_\alpha(G)\geq b_\alpha(n,k),$$

with equality if and only if $G=K_k\vee(K_1\cup K_{n-k-1})$.

Known cases in the paper include the bound for $\alpha\geq1$ and the related result at $\alpha=\tfrac12$. The conjecture asks for the complementary ranges $0<\alpha<1$ and $\alpha<0$, with the same extremal graph and equality characterization.