On graphs isomorphic with their conduction graph

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 GC 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 GC and G have the same vertex set, and each edge xy in GC represents a conducting device with graph G and connections x and y that conducts at the Fermi level. If GC is isomorphic with the simple graph G (in which case we call G conduction-isomorphic), then G has nullity η(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 η(G) = 0, GC is obtained by ‘booleanising’ the inverse adjacency matrix A−1(G), to form A(GC), i.e. by replacing all nonzero entries (A(G)−1)xy in the inverse by 1 + δxy where δxy 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 ≤ 11) connected conduction-isomorphic graphs and small (order ≤ 22) connected conduction-isomorphic graphs with maximum degree at most three is given. For η(G) = 1, it is shown that GC is connected if and only if G is a nut graph (a singular graph of nullity one that has a full kernel vector).