On Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph
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1 Aals of Pure ad Appled Mathematcs Vol. 3, No., 7, -3 ISSN: 79-87X (P, (ole Publshed o 3 March 7 DOI: Aals of O Eccetrcty Sum Egealue ad Eccetrcty Sum Eergy of a Graph D.S.Reakar, M.M.Patl ad H.S.Ramae 3 Departmet of Mathematcs KLE DR. M. S. Sheshger College of Egeerg & Techology Belgaum 98, Ida. Correspodg author. E-mal: reakards@redffmal.com Departmet of Mathematcs, Agad Isttute of Techology & Maagemet Belgaum 99, Ida. E-mal: maushamagdum@gmal.com 3 Departmet of Mathematcs, Karatak Uersty, Dharwad 83, Ida E-mal: hsramae@yahoo.com Receed February 7; accepted 8 February 7 Abstract. Let G be a smple graph wth ertces ad m edges. For a ertex ts eccetrcty, e s the largest dstace from to ay other ertces of G. I ths paper we troduce the cocept of eccetrcty sum matrx (G ad eccetrcty sum eergy E (G of a smple coected graph G ad obta bouds for egealues of (G ad bouds for the eccetrcty sum eergy E (G of a graph G. Keywords: Eccetrcty sum matrx, Egealues, Eccetrcty sum Eergy. AMS Mathematcs Subect Classfcato (: C. Itroducto Let G be a smple graph wth ertces ad m edges. Let the ertces of G be labeled as,,...,. The degree of a ertex a graph G, deoted by d( s the umber of edges cdet to. The dstace betwee the ertces ad s the legth of the shortest path og ad G. For a ertex ts eccetrcty, e s the largest dstace from to ay other ertces of G. The adacecy matrx A(G of a graph G s a square matrx of order whose (, -etry s equal to uty f the ertex s adacet to, ad s equal to zero otherwse. The egealues of adacecy matrx A(G are deoted by λ, λ,..., λ ad sce they are real t ca be ordered as λ λ λ. [] The eergy of a graph G s defed as [], E E ( G π Ths defto of eergy was motated by large umber of results for the Huckel molecular orbtal total π-electro eergy []. Motated by preous researches o Degree Sum Eergy of a Graph [3, ], ths paper we troduce eccetrcty sum matrx ad eccetrcty sum eergy assocated wth a graph ad study ts bouds. For more results o degree sum eergy see [8, 9] λ
2 D.S.Reakar, M.M.Patl ad H.S.Ramae 6 Let G be a smple graph wth ertces,,..., ad let e ecc( be the eccetrcty of,,,. The (G [a ] s called the eccetrcty sum matrx of a graph G where + otherwse, f, e e a ( The characterstc polyomal of the eccetrcty sum matrx s defed as, φ(g: det(i (G. where I s the detty matrx of order. Sce (G s real symmetrc matrx, the roots of φ(g: are real. These roots ca be ordered as, where s largest ad s smallest egealues. If G has,,, dstct egealues wth respecte multplctes k, k,, k the the spectrum ca be wrtte as, k k k G Spe L L ( The eccetrcty sum eergy of a graph G s defed as, G E ( ( Example. (G φ(g: ( + ( + 6 (. 9.88,, 3,.88, 6. Therefore, E (G Lemma.. [3] The Cauchy Schwarz equalty states that f (a, a,, a p ad (b, b,, b p are real p ectors the p p p b a a b. (3 Lemma.. [7] Let a, a,, a be o egate umbers. The Fgure : G : 3
3 O Eccetrcty Sum Egealue ad Eccetrcty Sum Eergy of a Graph a a / a ( a a. Bouds for the largest egealue of eccetrcty sum matrx Sce trace((g, the egealues of (G satsfes the relatos Further, trace ((G a a 7 a / ( ( a ( e + e < M where, M ( e + e ( < If r s the radus ad D s the dameter of a graph, the r e D. Hece, ( r M ( d. Equalty holds f r e D. Theorem.. If G s a coected graph wth ecc( e e,,,,, the G has oly oe poste egealue equal to ( e. Proof: Let G be a smple coected graph wth ertces. Let ecc( e e,,,,. The, e + e, f e,f a,otherwse,otherwse The the characterstc polyomal of the eccetrcty sum matrx s, φ(g: det(i (G det(i e (A(K. Where A(K s adacecy matrx of a complete graph K. det(i e (A(K ( e I A ( K (e ( + e e e [ ( e] ( + e Therefore [ ( e] ( + e. ( e e spect ( ( G. Hece G has oly oe poste egealue equal to ( e. Corollary.. If G K s a complete graph, the ( spe( ( K. Corollary.3. If G K p,q s a complete bpartte graph, the for p, q ( p + q spe( ( K p, q. p + q
4 Corollary.. If G C s a cycle, the D.S.Reakar, M.M.Patl ad H.S.Ramae ( spe( ( C, f s ee, ad spect ( ( ( ( C, f s odd. Corollary.. If G s a star graph S K,, the for ( ( spect S. Theorem.6. If G s ay graph wth ertces, the M ( (6 Equalty holds f ecc( e e,,,,. Proof: Let a ad b for, 3,, Eq. (3. Therefore From Eq. ( ad ( ad M. Therefore Eq. (7 becomes ( ( (M (. (7 whch ges, M (. For equalty, let ecc( e e,,,,. Therefore, M ( e + e e e ( e < < M( ( Hece ( e ( e. From Theorem., ( e s the oly oe poste egealue. Hece t s largest. Therefore equalty holds. 3. Bouds for the eccetrcty sum eergy of a graph Theorem 3.. If G s ay graph wth ertces, the M E ( G M. (8 Proof: Put a ad b from whch Eq. (3, we get 8
5 O Eccetrcty Sum Egealue ad Eccetrcty Sum Eergy of a Graph [E (G] (M E (G M. (9 Now, [E (G] M Therefore E ( G M ( Combg Eqs. (9 ad ( we get the result (8. Theorem 3.. Let G be ay graph wth ertces ad let be the absolute alue of the determat of the eccetrcty sum matrx (G. The M + ( E ( G ( M + ( Proof: Lower boud. By the defto of the eccetrcty sum eergy ad by Eq. ( [E (G] ( ( + M + ( < Sce for oegate umbers the arthmetc mea s ot smaller tha the geometrc mea, Therefore, ( ( ( ( /. Combg Eqs. ( ad (3 we get, [E (G] M + ( / (. (3.e. E (G M + ( ( Upper boud. Put,,,,. The from Lemma (. we obta, a / That s, M M [ E ( G] where, M ( Thus, [ E ( G] ( M + ( < ( e + e. Combg Equatos ( ad ( we obta the result ( Theorem 3.3. If G s a coected graph wth ecc( e e,,,,, the E (G ( e. Proof: If G s a coected graph wth ecc( e e,,,,, the from Theorem., G has oly oe poste egealue equal to ( e. Sce trace((g, sum of the remag egealues s equal to -( e. / 9
6 Therefore, D.S.Reakar, M.M.Patl ad H.S.Ramae E ( G ( e. Eccetrcty sum eergy of some graphs Graph G Eccetrcty Sum Eergy E (G K ( ( K m, ( 8( C (, f s ee (, f s ee (, f s odd (, f s odd. Coclusó For a coected graph G, we hae defed eccetrcty sum matrx (G ad eccetrcty sum eergy E (G. Show spectrum of some stadard graphs. Also obtaed bouds for egealues of (G ad bouds for the eccetrcty sum eergy E (G of a graph G. REFERENC. I.Gutma, The eergy of a graph, Ber. Math. Stat., 3 (978.. D.M.Cetkoc, M.Doob ad H.Sachs, Spectra of Graphs, Academc Press, New York, H.S.Ramae, D.S.Reakar ad J.B.Patl, Bouds for the degree sum egealues ad degree sum eergy of a graph, Iteratoal Joural of Pure ad Appled Mathematcal Sceces, 6 ( ( S.R.Jog, S.P.Hade ad D.S.Reakar, Degree sum polyomal of graph alued fuctos o regular graphs, It. J. Graph Theory, (3 (3 8.. S.Berard, J.M.Chld, Hgher Algebra, Macmlla Ida Ltd, New Delh,. 6. X.L, Y.Sh ad I.Gutma, Graph Eergy, Sprger, H.Kober, O the arthmetc ad geometrc meas ad o Holder's equalty, Proc. Amer. Math. Soc., 9 ( D.S.Reakar, M.M.Patl ad S.P.Hade, Note o the bouds for the degree sum eergy of a graph, degree sum eergy of a commo eghborhood graph ad termal dstace eergy of a graph, Iteratoal Joural of Mathematcal Arche, 7(6 ( S.M.Hosama ad H.S.Ramae, O degree sum eergy of a graph, Europea Joural of Pure ad Appled Mathematcs, 9(3 (
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