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Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSEMSE.

Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSE.

Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSE.

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Tony Huynh
  • 32.1k
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Is there any characterization of properties of two graphs GGiven a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and H such that there total graphs$b$ are isomorphic that is T(G) isomorphic to T(H)adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSE.

Is there any characterization of properties of two graphs G and H such that there total graphs are isomorphic that is T(G) isomorphic to T(H).

Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSE.

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sriram
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Characterization of non-isomorphic graphs but isomorphic total graphs?

Is there any characterization of properties of two graphs G and H such that there total graphs are isomorphic that is T(G) isomorphic to T(H).