how should I compute the homology groups of graphs for example the homology groups of $K5$ ? is it related to the genus of the graph?
2026-03-30 05:13:13.1774847593
how to compute the homology groups of graphs?
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A finite graph $G$ is a $1$-complex, so its $H_n$ vanish for $n\ge2$. If it is connected then $H_0(G,\Bbb Z)\cong\Bbb Z$. Then it is homotopy equivalent to a bouquet of circles, so $H_1(G,\Bbb Z)\cong\Bbb Z^a$ for some $a$. Its Euler characteristic is $1-a$, and that equals $v-e$ where $v$ and $e$ are the number of vertices and edges, respectively.
For non-connected graphs, you could compute the homology of each component.