2 problems

Let $H^{(v)}$ be an $\mathcal{F}$-preserver of order $m$, and let $G$ be any graph. Write $A(G)$ for the adjacency matrix of $G$, $W_A(G)$ for its adjacency walk matrix, and…

Let $H_{4t+1}$ be a graph of order $4t+1$, let $H_{4t+1}^{(\ell)}$ denote the rooted graph obtained by choosing root $\ell$, and let $G\circ H_{4t+1}^{(\ell)}$ be their rooted prod…