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).
