Guiduli–Mohar conjecture on minimum algebraic connectivity with prescribed minimum degree

Let $H$ be a connected graph of order $n$ with minimum degree $\delta(H)=d$, and let $\mu(H)$ denote the second-smallest eigenvalue of its Laplacian. A graph is $\mu$-minimal with minimum degree $d$ if it has the smallest algebraic connectivity among all such graphs.

Guiduli–Mohar conjecture. If $n\equiv0\pmod{d+1}$, then the $\mu$-minimal graph on $n$ vertices with $\delta=d$ is the graph displayed in Figure Mohar.

The conjecture predicts an explicit path-like block structure for extremal graphs; the cited figure specifies that structure, including complete-graph blocks. Its general status is not resolved in the supplied text.