On graphs isomorphic with their conduction graph

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 $G^{\prime }$ 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 $\le 11$) connected conduction-isomorphic graphs and small (order $\le 22$) 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).

Paper accepted for publication in MATCH Commun. Math. Comput. Chem.: https://doi.org/10.46793/match.93-2.379B