The extremal algebraic-connectivity conjecture for k-path graphs

Let $P_n^k$ denote the $k$-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 fixed $n\geq k+1$ and $k\geq 2$, the algebraic connectivity is the second-smallest Laplacian eigenvalue.

Extremal algebraic-connectivity conjecture. Given fixed $n\geq k+1$ and $k\geq 2$, the unique $k$-path graph that maximizes the algebraic connectivity is $K_{k-1}\vee P_{n-k+1}$. Moreover, under the same conditions, the unique $k$-path graph that minimizes the algebraic connectivity is $P_n^k$.

This conjecture identifies unique maximizers and minimizers of algebraic connectivity among $k$-path graphs. It is motivated by observed structural patterns and computational experiments; no resolution is supplied in the source.