On graphs isomorphic with their conduction graph

TL;DR

This paper introduces the concept of conduction graphs and explores conditions under which a graph is conduction-isomorphic to its conduction graph, revealing structural properties and providing classifications for small graphs.

Contribution

It defines conduction graphs in the context of molecular conductance, characterizes conduction-isomorphic graphs, and constructs examples and classifications, including answering a recent open question.

Findings

Conduction-isomorphic graphs have nullity zero and are ipso omni-insulators.

Constructed conduction-isomorphic graphs for graphs with minimum degree two or odd degree.

No 3-regular conduction-isomorphic graphs exist.

Abstract

Conduction graphs are defined here in order to elucidate at a glance the often complicated conduction behaviour of molecular graphs as ballistic molecular conductors. The graph $G^{\mathrm C}$ describes all possible conducting devices associated with a given base graph $G$ within the context of the Source-and-Sink-Potential model of ballistic conduction. The graphs $G^{\mathrm C}$ and $G$ have the same vertex set, and each edge $xy$ in $G^{\mathrm C}$ represents a conducting device with graph $G$ and connections $x$ and $y$ that conducts at the Fermi level. If $G^{\mathrm C}$ is isomorphic with the simple graph $G$ (in which case we call $G$ conduction-isomorphic), then $G$ has nullity $\eta(G)=0$ and is an ipso omni-insulator. Motivated by this, examples are provided of ipso omni-insulators of odd order, thereby answering a recent question. For $\eta(G)=0$, $G^{\mathrm C}$ is obtained by 'booleanising' the inverse adjacency matrix $A^{-1}(G)$, to form $A(G^{\mathrm C})$, i.e. by replacing all non-zero entries $(A(G)^{-1})_{xy}$ in the inverse by $1+\delta_{xy}$ where $\delta_{xy}$ is the Kronecker delta function. Constructions of conduction-isomorphic graphs are given for the cases of $G$ with minimum degree equal to two or any odd integer. Moreover, it is shown that given any connected non-bipartite conduction-isomorphic graph $G$, a larger conduction-isomorphic graph $G'$ with twice as many vertices and edges can be constructed. It is also shown that there are no 3-regular conduction-isomorphic graphs. A census of small (order $\leq 11$) connected conduction-isomorphic graphs and small (order $\leq 22$) connected conduction-isomorphic graphs with maximum degree at most three is given. For $\eta(G)=1$, it is shown that $G^{\mathrm C}$ is connected if and only if $G$ is a nut graph (a singular graph of nullity one that has a full kernel vector).