***************************************************************************** * 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: St000454 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(G): e = max([e for (e,_,_) in G.eigenvectors()]) if e in ZZ: return e ----------------------------------------------------------------------------- Statistic values: ([],1) => 0 ([],2) => 0 ([(0,1)],2) => 1 ([],3) => 0 ([(1,2)],3) => 1 ([(0,1),(0,2),(1,2)],3) => 2 ([],4) => 0 ([(2,3)],4) => 1 ([(0,3),(1,2)],4) => 1 ([(1,2),(1,3),(2,3)],4) => 2 ([(0,2),(0,3),(1,2),(1,3)],4) => 2 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3 ([],5) => 0 ([(3,4)],5) => 1 ([(0,4),(1,4),(2,4),(3,4)],5) => 2 ([(1,4),(2,3)],5) => 1 ([(2,3),(2,4),(3,4)],5) => 2 ([(1,3),(1,4),(2,3),(2,4)],5) => 2 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,1),(2,3),(2,4),(3,4)],5) => 2 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4 ([],6) => 0 ([(4,5)],6) => 1 ([(1,5),(2,5),(3,5),(4,5)],6) => 2 ([(2,5),(3,4)],6) => 1 ([(3,4),(3,5),(4,5)],6) => 2 ([(2,4),(2,5),(3,4),(3,5)],6) => 2 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,5),(1,4),(2,3)],6) => 1 ([(1,2),(3,4),(3,5),(4,5)],6) => 2 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 2 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 2 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 2 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 2 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 3 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 2 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 3 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(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) => 4 ([(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) => 5 ([],7) => 0 ([(5,6)],7) => 1 ([(2,6),(3,6),(4,6),(5,6)],7) => 2 ([(3,6),(4,5)],7) => 1 ([(4,5),(4,6),(5,6)],7) => 2 ([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 2 ([(3,5),(3,6),(4,5),(4,6)],7) => 2 ([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2 ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 2 ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(1,6),(2,5),(3,4)],7) => 1 ([(2,3),(4,5),(4,6),(5,6)],7) => 2 ([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 2 ([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => 2 ([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2 ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => 2 ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 3 ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 2 ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => 2 ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => 3 ([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => 2 ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2 ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => 3 ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 2 ([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => 2 ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => 2 ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7) => 2 ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 3 ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 3 ([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => 2 ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 4 ([(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) => 4 ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 3 ([(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) => 4 ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 3 ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4 ([(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) => 5 ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,6)],7) => 4 ([(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) => 5 ([(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) => 6 ----------------------------------------------------------------------------- Created: Apr 04, 2016 at 11:40 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 23, 2017 at 20:51 by Martin Rubey