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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.13518 |
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| _version_ | 1866909321081651200 |
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| author | Birkinshaw, Aidan Fowler, Patrick W. Goedgebeur, Jan Jooken, Jorik |
| author_facet | Birkinshaw, Aidan Fowler, Patrick W. Goedgebeur, Jan Jooken, Jorik |
| contents | 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 $η(G)=0$ 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 $η(G)=0$, $G^{\mathrm C}$ 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 $1+δ_{xy}$ where $δ_{xy}$ 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 $η(G)=1$, 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13518 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On graphs isomorphic with their conduction graph Birkinshaw, Aidan Fowler, Patrick W. Goedgebeur, Jan Jooken, Jorik Combinatorics 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 $η(G)=0$ 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 $η(G)=0$, $G^{\mathrm C}$ 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 $1+δ_{xy}$ where $δ_{xy}$ 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 $η(G)=1$, 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). |
| title | On graphs isomorphic with their conduction graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.13518 |