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 0-th group and the 1-th group.[1]
^Sunada, Toshikazu (2013), Sunada, Toshikazu (ed.), "Homology Groups of Graphs", Topological Crystallography: With a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, Tokyo: Springer Japan, pp. 37–51, doi:10.1007/978-4-431-54177-6_4, ISBN 978-4-431-54177-6
In algebraic topology and graph theory, graphhomology describes the homology groups of a graph, where the graph is considered as a topological space....
invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. The Euler characteristic χ was...
they are all finite. The nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximum number of cuts that can be...
included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain...
of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory,...
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises...
Sequence graph, also called an alignment graph, breakpoint graph, or adjacency graph, are bidirected graphs used in comparative genomics. The structure...
holes, of a topological space. In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure...
its complement graph, and vice versa. Several authors studied the relations between the properties of a graph G = (V, E), and the homology groups of its...
{rank} H_{1}(G,t),} which is read as "the rank of the first homology group of the graph G relative to the terminal nodes t". This is a technical way...
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins...
complexes and cellular maps, cellular homology can be interpreted as a homology theory. To compute an extraordinary (co)homology theory for a CW complex, the Atiyah–Hirzebruch...
combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory...
its 0-th reduced homology group is trivial: H 0 ~ ( X ) ≅ 0 {\displaystyle {\tilde {H_{0}}}(X)\cong 0} . For example, when X is a graph and its set of connected...
Reeb graph is related to Morse theory and MAPPER is derived from it. The proof of this theorem relies on the interleaving distance. Persistent homology is...
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is in D, or has a neighbor in D. The domination...
function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings, and...
mathematics: A graph without a cycle, especially A directed acyclic graph An acyclic complex is a chain complex all of whose homology groups are zero...
deformation quantization of Poisson manifolds, the Deligne conjecture, or graphhomology in the work of Maxim Kontsevich and Thomas Willwacher. Suppose X {\displaystyle...
of independent cycles can also be explained using homology theory, a branch of topology. Any graph G may be viewed as an example of a 1-dimensional simplicial...
for each vertex of the graph and a line segment for each edge of the graph. This construction may be generalized to the homology group H 1 ( G , R ) {\displaystyle...
theory Graph theory Grothendieck's Galois theory Group theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Invariant...
its homotopy groups. Homological connectivity, a property related to the homology groups of a topological space. Homotopic connectivity - connectivity between...
polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Peter Ozsváth and Zoltan Szabó is the...
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