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I am studying a special kind of graphs, and I would like to know if they are studied in the literature and what they are called. Let $G$ be a simple, finite, undirected, connected graph, with vertex set $V$ and edge set $E$. Consider the $\mathbb R$-vector space that is generated by all pairs $(x,y)\in V^2$ with $\{x,y\}\in E$, using the convention that $(x,y) = -(y,x)$. This vector space is isomorphic to $\mathbb C[E]$, choosing a direction for each edge. To each cycle $C = (x_1,\ldots,x_n)$ in the graph we associate the element $(x_1,x_2)+(x_2,x_3)+\ldots+(x_{n-1},x_n)+(x_n,x_1)$ in this vector space. The subspace generated by all cycles is denoted $Z(G)$. It is characterized by the equality $\sum_{y\sim x}c((x,y)) = 0$ for all $c\in Z(G)$ and $x\in V$.

In particular, to any triangle $(x_1,x_2,x_3)$ in the graph we associate the cycle $(x_1,x_2)+(x_2,x_3)+(x_3,x_1)$. The vector space generated by all triangles is denoted $Z_3(G) \subseteq Z(G)$. The graphs I am interested in are the graphs for which $Z_3(G) = Z(G)$. Equivalently, there should be $|E|-|V|+1$ linearly independent triangles. A direct consequence is that every edge that is an cycle is also in a triangle. Examples of these graphs are triangular grids and wheel graphs. Every chordal graph satisfies $Z_3(G) = Z(G)$ but the opposite is not true.

Is this kind of graph studied in the literature and what is it called?

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    $\begingroup$ basically you are considering simplicial complexes of vertices, edges, and triangles of graphs. A complex is called simply connected if its fundamental group F is trivial. Your condition says that F is perfect, i.e. its abelianisation is trivial. $\endgroup$ Commented Jul 22, 2022 at 15:55
  • $\begingroup$ What's an example of a graph that satisfies the condition but isn't chordal? $\endgroup$ Commented Jul 22, 2022 at 23:40
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    $\begingroup$ A wheel graph is not chordal. $\endgroup$ Commented Jul 23, 2022 at 8:03
  • $\begingroup$ Do connected graphs whose clique complexes are simply connected satisfy the condition? $\endgroup$ Commented Jul 23, 2022 at 17:23
  • $\begingroup$ I think so yes. $\endgroup$ Commented Jul 26, 2022 at 13:30

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We have recently posted a preprint on Arxiv that collects/proves several sufficient graph theoretic conditions for this to hold, with particular focus on strongly regular graphs. They appear in section 5 of the paper: https://arxiv.org/abs/2508.05871.

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