9 problems

Let $\operatorname{\mathsf{C}}_{n}$ denote the cycle graph on $n$ vertices. A graph is $\mathcal{L}$-DS when its normalized Laplacian spectrum determines it up to isomorphism. Butl…

Normalized algebraic-connectivity conjecture.

Let $G$ be a graph on $n$ vertices, and let $\lambda_2(G)$ denote the second-smallest eigenvalue of its normalized Laplacian. Connected-complement normalized Laplacian conjecture.…

Let $G$ be a graph on $n\geq5$ vertices, with normalized Laplacian eigenvalues $0=\lambda_1(G)\leq\lambda_2(G)\leq\cdots\leq\lambda_n(G)$. Normalized Laplacian maximum-bound conjec…

Normalized Laplacian maximum-connectivity conjecture.

Let $G$ be a graph, let $x,y$ be vertices, and let $\lambda$ be an eigenvalue of the normalized Laplacian whose edge derivative is being considered. The edge derivative with respec…

Normalized-Laplacian tree state-transfer conjecture. Apart from the paths on $2$ and $3$ vertices, no tree admits perfect state transfer according to the normalized Laplacian.

Let $G$ be a graph with $\rho_{n-1}(G)\neq 1$, matching number $\nu(G)=2$, and diameter $\operatorname{diam}(G)=2$; these are the remaining graphs to consider in the characterizati…

Let $\mathcal{G}(n,n-3)$ denote the class of graphs considered in the paper, namely graphs on $n$ vertices whose normalized Laplacian eigenvalue $1$ has multiplicity $n-3$. In the…