23 problems

Resistance-distance conjecture. The resistance distance is bounded above by the directed distance:

Let $G$ and $H$ be graphs, and let $\operatorname{RS}(G)$ denote the multiset of resistance distances between all pairs of distinct vertices of $G$. Baxter's conjecture. Two graphs…

Let $K_{m,n}$ be the complete bipartite graph whose vertex set is partitioned into parts of sizes $m$ and $n$, with every vertex in one part adjacent to every vertex in the other.…

Let $G$ be the straight linear 3-tree with $n$ vertices, and let $H$ be the straight linear 3-tree with $n+1$ vertices. Write $r_G(1,n)$ and $r_H(1,n+1)$ for the total resistance b…

Let $G$ be the straight linear $3$-tree with $n$ vertices and $H$ the straight linear $3$-tree with $n+1$ vertices. For vertices $u$ and $v$, let $r_G(u,v)$ and $r_H(u,v)$ denote t…

Let $G$ be a connected graph on $n\geq 3$ vertices. For each vertex $v_i$, let $\kappa_i$ be its resistance curvature, defined by the unique solution $\kappa$ of … where $R_G$ is t…

Let a family of graphs have Laplacian matrices, and apply the procedures LaplaceExpand and SystemReduce together with the mathematical theorems described in the source. The numerat…

Let $G$ be the straight linear $3$-tree with $n$ vertices, and let $H$ be the straight linear $3$-tree with $n+1$ vertices. Write $r_G(1,n)$ and $r_H(1,n+1)$ for the total resistan…

Let $G$ be a connected graph. The resistance distance $R_G[u,v]$ between vertices $u$ and $v$ is the net effective resistance between them when each edge of $G$ is replaced by a un…

Let $T_n$ be the triangular grid graph shown in Figure, with $n$ rows and $m=n^2$ cells. Let $a$ and $b$ be distinct vertices of degree $2$, and let $r_n(a,b)$ be the resistance di…

Let $G=P_n\square P_n$ be the $n\times n$ grid graph with $n^2$ vertices, and let $H=P_{n+1}\square P_{n+1}$ be the $(n+1)\times(n+1)$ grid graph with $(n+1)^2$ vertices. Let…

Let $G$ be the block tower graph $C_4\square P_n$, the Cartesian product of the $4$-cycle and the path of length $n$, with $4n$ vertices, and let $H$ be the block tower graph…

Let $G$ be the straight linear $K$-tree with $n$ vertices and parameter $k\geq 1$, and let $H$ be the straight linear $K$-tree with $n+1$ vertices. Write $r_G(u,v)$ for the resista…

A weighted $k$-nested network is a weighted network in which the relevant nested structure has depth $k$, and its resistance distance is the pairwise effective resistance between v…

Kirchhoff-index ordering conjecture. For $\alpha\in\{-1,1\}$,

Exterior-shape invariance conjecture. If

2-nested network conjecture. The resistance distance of every 2-nested phylogenetic network is Kalmanson.

Realizability conjecture. If $d53d$ is faithfully phylogenetic, then there exists a weighted phylogenetic network $N$ such that

Outer planarity conjecture. Outer planarity is sufficient for the resistance distance to be Kalmanson.

Let $T_n$ be the triangular grid graph with $n$ rows and $m=n^2$ cells. Let $a$ and $b$ be distinct degree-two vertices of $T_n$, and let $r_n(a,b)$ be their resistance distance. T…

Let $G$ be a linear 2-tree, and let its diameter tend to infinity. Let the maximal resistance distance of $G$ mean the maximum of $r_G(u,v)$ over pairs of vertices. Unbounded resis…

Let $G$ be the straight linear $k$-tree with $k\geq 1$ and $n$ vertices, and let $H$ be the straight linear $k$-tree with $n+1$ vertices. Let $r_G$ and $r_H$ denote resistance dist…

Let $G_n$ be the straight linear $k$-tree with vertex set $\{1,2,\ldots,n\}$ and edges between vertices whose labels differ by at most $k$. Similar asymptotic behavior should hold…