Let $G$ be a finite connected simple graph, let $uv$ be a nonedge, and let $s$ be a starting vertex. We study the exact change in the expected cover time of simple random walk when $uv$ is inserted. A killed Green matrix update, combined with target-set inclusion-exclusion, gives an exact formula using only the original graph. The response can have either sign. Our main result is a nonlocality theorem. For every radius $r\geq 1$, we construct two marked configurations whose ambient-degree-labelled radius-$r$ neighbourhoods at $s,u,v$ are isomorphic but whose fixed-start cover-time responses have opposite signs. The construction also matches the three marked degrees, the marked distance, and the effective resistance $R_{uv}$. The two graphs have different orders; equal-order nonlocality remains open. We complement this result with a conductance interpolation theorem and an occupation interpretation of its high-conductance coefficient. After contracting $u$ and $v$, that coefficient is a positive multiple of the expected pre-cover occupation of the contracted vertex, and its zero case is classified exactly. A path with two pendant insertion endpoints is solved for all starting vertices and shows a sharp change of sign across the start set.
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $\mu_c(G)$, defined as the maximum cardinality o...
Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\ge2$ vertices and $e$ edges, we prove $\operatorname{Var}(X_G)\le e((n-2)/n)^2$, with...
We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings betwee...
Let $\mathcal P_1$ denote the class of 1-planar graphs and let $tC_5$ be the disjoint union of $t$ copies of $C_5$. For every fixed $t\ge3$ and all sufficiently large $n$, we determine the unique $n$-vertex $tC_5$-free graph in $\mathcal P_1$ with maximum adjacency spectral radius, answering Problem 1 of Li, Wang and Z...
Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry--\'Emery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We pro...
