Defines the counting matrix K of a simplicial complex and establishes that it lies in SL(n,Z) with explicit inverse, positive definiteness, and spectral symmetry between K and its inverse.
The energy of a simplicial complex
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abstract
A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy summing all matrix elements g(x,y) is equal to the Euler characteristic X(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to X(G).
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math.CO 1years
2019 1verdicts
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The counting matrix of a simplicial complex
Defines the counting matrix K of a simplicial complex and establishes that it lies in SL(n,Z) with explicit inverse, positive definiteness, and spectral symmetry between K and its inverse.