In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1-complex (i.e., its 'faces' are the vertices – which are 0-dimensional, and the edges – which are 1-dimensional), the only non-trivial homology groups are the 0th group and the 1st group.
1st homology group The general formula for the 1st homology group of a topological space X is: H 1 ( X ) := ker ∂ 1 / im ∂ 2 {\displaystyle H_{1}(X):=\ker \partial _{1}{\big /}\operatorname {im} \partial _{2}} The example below explains these symbols and concepts in full detail on a graph.
Example Let X be a directed graph with 3 vertices {x, y, z} and 4 edges {a: x → y, b: y → z, c: z → x, d: z → x}. It has several cycles:
One cycle is represented by the loop a+b+c. Here, the plus sign represents that all edges are travelled at the same direction. Since the addition operation is commutative, the + sign represents that the loops a + b + c, b + c + a, and c + a + b, all represent the same cycle. A second cycle is represented by the loop a + b + d. A third cycle is represented by the loop c − d. Here, the minus sign represents that the edge d is travelled backwards. If we cut the plane along the loop a + b + d, and then cut at c and "glue" at d, we get a cut along the loop a + b + c. This can be represented by the following relation: (a + b + d) + (c − d) = (a + b + c). To formally define this relation, we define the following commutative groups:
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