The extremal alpha-index and algebraic-connectivity conjecture for k-path graphs

Let $P_n^k$ denote a $k$-path graph of order $n$, let $K_{k-1}$ be the complete graph on $k-1$ vertices, let $P_{n-k+1}$ be the path on $n-k+1$ vertices, and let $\vee$ denote the graph join. For a graph $G$, write $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$ for $\alpha\in[0,1]$, where $A(G)$ and $D(G)$ are its adjacency and degree matrices; the $\alpha$-index is the largest eigenvalue of $A_\alpha(G)$. The $k$-weak-generalized-fan is the unique $k$-path graph, up to isomorphism, having a vertex $v$ such that deleting $v$ yields $K_{k-1}\vee P_{n-k}$.

Extremal alpha-index and algebraic-connectivity conjecture. Given fixed $n\geq k+1$ and $k\geq 2$, the unique $k$-path graph that maximizes the $\alpha$-index for $\alpha\in[0,1]$ is $K_{k-1}\vee P_{n-k+1}$. Moreover, under the same conditions, the unique $k$-path graph that maximizes $\lambda_2(A_\alpha)$ is the $k$-weak-generalized-fan.

The conjecture proposes simultaneous extremal descriptions for the $\alpha$-index and for the second eigenvalue of $A_\alpha$. It is based on the structural definitions and experiments presented in the source; no proof or resolution is given there.