***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000422 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph \$K_n\$ equals \$2n-2\$. For this reason, we do not define the energy of the empty graph. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Graph_energy]] [2] Bapat, R. B., Pati, S. Energy of a graph is never an odd integer [[MathSciNet:2250987]] [3] Pirzada, S., Gutman, I. Energy of a graph is never the square root of an odd integer [[MathSciNet:2396734]] ----------------------------------------------------------------------------- Code: def statistic(G): return sum(abs(c) for c in G.spectrum()) ----------------------------------------------------------------------------- Statistic values: ([],1) => 0 ([],2) => 0 ([(0,1)],2) => 2 ([],3) => 0 ([(1,2)],3) => 2 ([(0,1),(0,2),(1,2)],3) => 4 ([],4) => 0 ([(2,3)],4) => 2 ([(0,3),(1,2)],4) => 4 ([(1,2),(1,3),(2,3)],4) => 4 ([(0,2),(0,3),(1,2),(1,3)],4) => 4 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6 ([],5) => 0 ([(3,4)],5) => 2 ([(0,4),(1,4),(2,4),(3,4)],5) => 4 ([(1,4),(2,3)],5) => 4 ([(2,3),(2,4),(3,4)],5) => 4 ([(1,3),(1,4),(2,3),(2,4)],5) => 4 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6 ([(0,1),(2,3),(2,4),(3,4)],5) => 6 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8 ([],6) => 0 ([(4,5)],6) => 2 ([(1,5),(2,5),(3,5),(4,5)],6) => 4 ([(2,5),(3,4)],6) => 4 ([(3,4),(3,5),(4,5)],6) => 4 ([(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3)],6) => 6 ([(1,2),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 8 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 8 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 8 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([],7) => 0 ([(5,6)],7) => 2 ([(2,6),(3,6),(4,6),(5,6)],7) => 4 ([(3,6),(4,5)],7) => 4 ([(4,5),(4,6),(5,6)],7) => 4 ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 6 ([(3,5),(3,6),(4,5),(4,6)],7) => 4 ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 6 ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(1,6),(2,5),(3,4)],7) => 6 ([(2,3),(4,5),(4,6),(5,6)],7) => 6 ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 6 ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6 ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 8 ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 8 ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 6 ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 8 ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8 ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 10 ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 8 ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 8 ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 8 ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => 8 ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 8 ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => 8 ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 10 ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 10 ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 10 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 10 ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 10 ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 12 ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 10 ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 12 ----------------------------------------------------------------------------- Created: Feb 26, 2016 at 11:47 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 26, 2016 at 12:08 by Martin Rubey