The Eigen-Cover Ratio of a Graph: Asymptotes, Domination and Areas

Transcription

1 The ige-cover Ratio of a Graph: Asymptotes, Domiatio ad Areas Paul August Witer ad Carol Lye Jessop Mathematics, UKZN, Durba, outh Africa- witerp@ukzacza Abstract The separate study of the two cocepts of eergy ad vertex coverigs of graphs has opeed may aveues of research I this paper we combie these two cocepts i a ratio, called the eige-cover ratio, to ivestigate the domiatio effect of the subgraph iduced by a vertex coverig of a graph G called the cover graph of G ), o the origial eergy of G, where large umber of vertices are ivolved This is referred to as the eige-cover domiatio ad has relevace, i terms of coservatio of eergy, whe a molecule s atoms ad bods are mapped oto a graph with vertices ad edges, respectively If this eergy-cover ratio is a fuctio of, the order of graphs belogig to a class of graph, the we discuss its horizotal asymptotic behavior ad attach the graphs average degree to the Riema itegral of this ratio, thus associatig eige-cover area with classes of graphs We foud that the eige-cover domiatio had a strogest effect o the complete graph, while this chromatic-cover domiatio had zero effect o star graphs We show that the eige-cover asymptote of discussed classes of graphs belog to the iterval [0,], ad cojecture that the class of complete graphs has the largest eige-cover area of all classes of graphs AM classificatio 05C99 ORCID: Key words: eergy of graphs, eigevalues, vertex cover, domiatio, ratios, asymptotes, areas

2 Itroductio All graphs i this paper are simple ad loopless ad o vertices ad m edges We shall use the graph-theoretical otatio of Harris, Hirst ad Mossighoff [8] ergy, vertex covers ad ratios of graphs Much research has bee doe ivolvig the eergy of a graph see Adiga, Bayad, Gutma ad riivas [] ad Coulso, O'Leary ad Mallio []) ad miimum) vertex coverigs of graph see Adiga, Bayad, Gutma ad riivas []) Ratios have bee a importat aspect of graph theoretical defiitios xamples of ratios are: expaders see Alo ad pecer []), the cetral ratio of a graph see Buckley [3]), eige-pair ratio of classes of graphs see Witer ad Jessop [3]), Idepedece ad Hall ratios see Gábor []), tree-cover ratio of graphs see Witer ad Adewusi []), the eige-eergy formatio ratio of graphs see Witer ad arvate [6] ), the t-complete eige ratio of graphs see Witer, Jessop ad Adewusi []), the chromatic-cover ratio of graphs see Witer [9]), the chromaticcomplete differece ratio of graphs see Witer [0], the tree-3-cover ratio of graphs see Witer []) ad the eige-complete differece ratio see Witer ad Ojako [5]), I this paper we combie the two cocepts of eergy ad vertex coverig of graphs to form a ratio, the eige-cover ratio, associated with a coected graph G, ivolvig the eergy of the sugraph H of iduced by a vertex cover of G, called the cover graph of G, ad the eergy of G This eige-cover ratio allows for the ivestigatio of the domiatio effect of the eergy of the cover graph o the origial eergy of G, where a large umber of vertices are ivolved referred to as the eige-cover domiatio This eige-cover ratio has relevace whe molecules with atoms ad bods are mapped oto a graph with vertices ad edges, respectively I terms of coservatio of eergy, oe requires the smallest set of atoms whose excitatio would affect all atoms which ca be reached from by a path of legth at most, ad where a large amout of atoms are ivolved the eige-domiatio effect becomes relevat If the eige-cover ratio is a fuctio of, the order of graphs belogig to a particular class of graphs, the we ivestigated its asymptotic behavior see Witer ad Adewusi [],Witer ad G

3 3 Jessop [3], Witer, Jessop ad Adewusi [], Witer ad arvate [6],Witer ad Ojako [5], Witer[9],[0], ad[],) The eige-cover domiatio was determied for kow classes of graph We foud that, for the complete graph, the eige-cover domiatio was the strogest for complete graphs, ad for star graphs with rays of legth oe, o effect at all By itroducig the average degree of a graph together with the Riema itegral of the eige-cover ratio we associated eige cover area with classes of graphs see Witer ad Adewusi[], Witer ad Jessop[3], Witer ad arvate[6], Witer, Jessop ad Adewusi [],Witer ad Ojako[5], Witer[9],[0], ad[]) ige-cover ratio, asymptotes, domiatio ad area We combie the idea of eergy defied below) ad vertex cover the miimum set of vertices of a graph G such that every edge of G has at least oe ed i ) of graphs i the followig defiitios, to allow for the measure of the domiatio of the eergy of a cover graph over the eergy of origial graph for large values of If oe cosiders a molecule i a graph-theoretical way, where the atoms are the vertices ad the edges the bods betwee the atoms, the the idea of eergizig the whole molecule is relevat Coservig eergy will ivolve the smallest set of atoms which ca be eergized so that all atoms outside, ad coected to by a path of legth at most oe, also be eergized This is equivalet, graphically, to fidig a miimum vertex coverig of a graph, ie every path of legth has at least oe ed i see Adiga, Bayad, Gutma ad riivas []) Whe a large umber of atoms are ivolved the asymptotic behavior of this eige-cover ratio becomes sigificat Defiitio The Huckel Molecular Orbital theory provided the motivatio for the idea of the eergy of a graph - the sum of the absolute values of the eigevalues associated with the graph Adiga, Bayad, Gutma ad riivas [] ad Coulso, O'Leary ad Mallio []): i, where i, i are eigevalues of adjacecy matrix of graph G

4 The defiitios below allows for the ivestigatio of the eergy-domiatio of, ad the eige-cover areas, of classes of graphs see Witer ad Adewusi [], Witer ad Jessop [3], Witer ad arvate [6], Witer, Jessop ad Adewusi [], Witer ad Ojako[5], Witer[9],[0], ad[] for similar defiitios) Defiitio Let G be a coected graph with miimum coverig subgraph of G iduced by, be the cover graph of G of vertices Let H, the The eige-cover ratio of a graph G of order, with respect to, is defied as: G cov{ )} H )) where is the eergy of G oly the we defie H)) as Defiitio 3 defied above If H) cosists of isolated vertices If cov{ } f ) for every G, where is a class of graphs, the the asymptotic behavior of f is called the eige-cover asymptote of ad deoted by: as cov{ )} ige cover domiatio This asymptote give a measure of the domiatio effect of the eergy of the cover graph o the eergy of the origial graph, for large values of, referred to as the eige-cover domiatio Defiitio If cov{ } f ) for every G, where is a class of graphs,the the eige-cover area is defied as : m ) f ) d A

5 5 with A k ) 0 where k cov{ } f ) is defied, ad is the smallest umber of vertices for which m is the average degree of G, referred to as the legth of G while the itegral part is its height which we always make positive xamples: The complete graph K For the complete graph the complete graph is K, we have its cover graph H ) K The eergy of K ) i ad therefore the eergy of its cover graph is H )) K ) Hece, cov{ as cov{ K H )) ) ) ) ) )} ad ) ) K )} m The A K f ) d ) d ) l c) Now, A K 0 c l, so A K ) l l )

6 6 The complete split-bipartite graph K, For the complete split-bipartite graph partite sets o zero eergy Therefore, K,, we have vertices, ad its cover graph the set of cosistig of oe of the isolated vertices with cov{ H )) )} ad K, as cov{ K, )} 0 The eergy of G is m so that A [ d] l c) K ad, A K, 0 c l Hece the eige-cover area of the complete-split bipartite graph is or 3 The cycle graph C For the cycle graph C o a eve umber of vertices,we have with havig size by cosiderig every secod vertex of the cycle so that the cover graph cosists j of isolated vertices The eergy of the cycle is: ) cos 0

7 7 cov{ C )} H )) 0 j cos ) A C 0 j cos ) d l c) The cycle o a odd umber of vertices We take as a pair of adjacet vertices ad the alteratig vertices The triagle will have just the two adjacet vertices The cycle o five vertices will have two adjacet vertices ad a isolated vertex Geerally will cosist of 3 vertices ad will have eergy cov{ C )} H )) C ) ) ) C) C) j0 ) j cos ) A C m f ) d d l c 5 The path graph P For the path graph P o a eve umber of vertices, we have havig size by cosiderig the first vertex of the path ad the skippig a vertex so that the cover graph cosists of isolated vertices The eergy of the path graph is: cos j

8 8 j P P P H cos ) )} cov{ ) )) ) l c m A P

9 9 6 Path graphs o a odd umber of vertices Let P be a path o a odd umber of vertices will cosist of the secod vertex ad the skippig every vertex so that will be o vertices cov{ P )} H )) P ) P ) cos j A P m dv d l c) cos j 7 The wheel W, with spokes, ad with odd For the wheel graph W, with spokes ad where vertex ad every secod vertex of the cycle as the vertices i The eergy of the wheel graph is: is odd, we have the cetral W ) k k cos k k cos ) Ad, the cover graph is the star graph with rays of legth

10 0 Figure 7: Wheel graph W 9 Figure 7: Cover graph of W 9, H The eergy of H is 0 H i i )) ) ) cos ) )} cov{ ) )) k W W k W H 0 )} cov{ W as )) ) ) cos ) ) d d k d f m A k W

11 8 tar graphs r,o vertices with r rays of legth Figure 8: tar graph 8, The vertex cover comprises of the cetral vertex oly The cov{ r, )} H )) r,) as cov{ r, )} 0 A r, m ) f ) d d sec x d sec x taxdx ) ) dx arcsec c) A, 0 c arc sec

12 9 tar graphs r, with r rays of legth We have r, ad cover graph cosists of cosists of the middle vertex of each ray, so that the r isolated vertices The eigevalues of r, are : ad -, each of multiplicity eigevalue 0, ad two eigevalues r 3 r, oe The eergy of this graph is therefore: 3 The Figure 9: tar graph r, cov{ r, )} H )) r, ) 3 ) 3 )

13 3 as cov{ r, )} 0 m A f ) d r, ) 3 d ) 0 Lollipop graph Lemma Let G be a graph with a ed vertex subgraph of G iduced by removig the vertex x adjacet to vertex x ad let iduced by removig the vertex x The see Haemers [7]): P ) PA G') ) PA G'' ) A ) x, ad let Where P ) is the characteristic polyomial det A I) A xample with complete graph joied to ed vertex G' be the G'' be the subgraph of G o if G LP is the complete graph F o - vertices joied to a sigle ed vertex by a edge x x, we have: x P 3 A ) PA G') ) PA G'') ) ) )) ) 3)) ) ) ) [ ) ) 3))] [ ) ) 3)] [ ) ] Roots of quadratic are:

14 ) ; we have roots: 0; multipliti cy ) 3); 8 ) ; 8 ergy of this graph is therefore: 0 3) ) 8 8 ) sice 3) 8 Theorem The eergy of the lollipop graph is: LP ) 3) 8 A cover graph is the subgraph iduced by - vertices of the complete graph F icludig Hece: x so that its eergy is -6 cov{ LP )} H )) r,) ) 6) [ 3) 8] For large this behaves like: ) as cov{ LP )} m A [ ) ) ] LP f ) d ) 6) d [ 3) 8]

15 5 Dual star graph A dual star Du is defied as two star graphs with m rays of legth each o vertices) joied by a edge its ceter edge) coectig their ceters This graph has o-zero eigevalues foud as solutios of the followig equatio see Haemers [6]): x m ) x m 0 x ) x x x ) ) ) 3 ) or ) ) ) 3 3 Thus ) 3 ) 3 Du ) A cover set cosists of the vertices of the ceter edge of the graph Hece: cov{ Du )} H )) [ ) 3 ) 3 ] Ad as cov{ Du )} 0

16 6 A Du ) ) f ) d [ ) 3 ) d 3 ] For large this ratio behaves like: 3 [ 3 Thus for large the area associated with this ratio is: A Du 3 m [ d] ) c); large Makig the height aspect positive we have: A Du ) c); A Du 0 c Theorem The eige-cover ratio, asymptote ad area respectively for the followig classes of graphs are:

17 7 K : ; ; ) l l ) ad K, : ;0 ; l l ) ad 3 C : j0 j cos ;-; l c) eve C : j0 ) j cos ) ;-; l c odd 5 6 P P : : cos j ;-; l c) eve cos j ;-; l c) 7 W : ) k k cos ) ) d ) ) ;0; )) 8 r, : ) ) dx arcsec arcs sec ;0 ;) 9 r, : 3 ) ; 0 ; ) d 3 )

18 8 ) 6) [ ) ) ] ) 6) 0 LP : ;; d [ 3) 8] [ 3) 8] Du : ;0 ; ) 3 ) 3 [ ] ) [ ) 3 ) d 3 ] A Du ) ); large Corollary H )) If cov{ } f ) for each G, the as cov{ )} [0,] for all classes of graphs examied i the theorem above Cojecture The complete graph possesses the largest eige-cover area of all classes of graphs 3 Coclusio I this paper we combied the cocepts of eergy ad vertex coverig of Gthe coverig graph is the subgraph H) of G iduced by ) to determie the

19 9 domiatio effect of the eergy of the coverig graph o the mai graph G This ivolved the eige-cover ratio: G H )) cov{ )} Regardig a molecule i a graph-theoretical way, where the atoms are the vertices ad the edges the bods betwee the atoms, the the idea of eergizig the whole molecule is relevat Coservig eergy will ivolve the smallest set of atoms which ca be eergized so that all atoms outside, ad coected to by a path of legth at most oe, also be eergized This is equivalet, graphically, to fidig a miimum vertex coverig of a graph Whe a large umber of atoms are ivolved the asymptotic behavior of this eige-cover ratio becomes sigificat as illustrated by complete graphs havig the strogest eige-cover domiatio, implyig that activatio of its vertex cover will result i immediate activatio of all atoms whe a large umber of atoms are ivolved If this eige-cover ratio is a fuctio of, the order of G belogig to a class of graphs, the we determied the horizotal asymptote of the ratio, ad by attachig the average degree of G to the Riema itegral we foud the eige-cover area of classes of graphs We foud that the eige-cover asymptote value the domiatio effect) for classes of graphs ivestigated belogs to the iterval [0,] ad claim that the eige-cover area is greatest of all eige-cover areas of classes of graphs Refereces [] Adiga, C Bayad, A Gutma, I ad riivas, A The miimum coverig eergy of a graph Kragujevac J ci 3, ) [] Alo, N ad pecer, J H 0 igevalues ad xpaders The Probabilistic Method 3rd ed) Joh Wiley & os [3] Buckley, F 98 The cetral ratio of a graph Discrete Mathematics 38): 7

20 0 [] Coulso, C A O'Leary, B ad Mallio, R B 978 Hückel Theory for Orgaic Chemists Academic Press [5] Gábor, 006 Asymptotic values of the Hall-ratio for graph powers Discrete Mathematics3069 0): [6] Haemers, W H mall itegral trees wwwwituel/~aeb/graphs/itegral_treeshtml [7] Haemers, W H, Liu, X ad Zhag, Y 008 pectral characterizatios of lollipop graphs Liear Algebra ad its Applicatios8 ), 5 3 [8] Harris, J M, Hirst, J L ad Mossighoff, M 008 Combiatorics ad Graph theory priger, New York [9] Witer, P A 05 The chromatic-cover ratio of a graph: domiatio, areas ad farey sequeces To appear i the Iteratioal Joural of Mathematical Aalysis [0] Witer, P A 05 The chromatic-complete differece ratio of graphs Vixra, 3 pages [] Witer, P A 05 Tree-3-cover ratio of graphs: asymptotes ad areas vixra: pages [] Witer, P A ad Adewusi, FJ 0 Tree-cover ratio of graphs with asymptotic covergece idetical to the secretary problem Advaces i Mathematics: cietific Joural; Volume 3, issue, 7-6 [3] Witer, P A ad Jessop, CL 0Itegral eige-pair balaced classes of graphs with their ratio, asymptote, area ad ivolutio complemetary aspects Iteratioal Joural of Combiatorics Volume 0 Article ID 8690, 6 pages

21 [] Witer, P A, Jessop, C L ad Adewusi, F J 05 The complete graph: eigevalues, trigoometrical uit-equatios with associated t-complete-eige sequeces, ratios, sums ad diagrams To appear i Joural of Mathematics ad ystem ciece [5] Witer, P A ad Ojako, O 05 The eige-complete differece ratio of classes of graphs- domiatio, asymptotes ad area 05 Vixraorg, To appear i Joural of Advaces i Mathematics [6] Witer, P A ad arvate, D 0 The h-eige eergy formatio umber of h-decomposable classes of graphs- formatio ratios, asymptotes ad power Advaces i Mathematics: cietific Joural; Volume 3, issue, 33-7