2019/07/07 by Oliver Knill, Knill, Oliver · 1 citation
Computer Science · Mathematics · #05C10 #57M15 #68R10 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO #msc:05C10 #msc:57M15 #msc:68R10
paper · pdf · doi:10.48550/arxiv.1907.03369
34 pages
arxiv created 2019/07/07 · arxiv updated 2019/07/09
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).