Feng, Lihua

Liu, Weijun

Yu, Guihai

Repository of the Academy's Library

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No 1, pp. 97-108 DOI: 10.18514/MMN.2014.781Further results regarding the degree Kirchhoﬀindex of graphsLihua Feng, Guihai Yu, and Weijun LiuMiskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 15 (2014), No. 1, pp. 97–108FURTHER RESULTS REGARDING THE DEGREE KIRCHHOFFINDEX OF GRAPHSLIHUA FENG, GUIHAI YU, AND WEIJUN LIUReceived 24 September, 2013Abstract. Let G be a connected graph with vertex set V.G/. The degree Kirchhoff index of Gis defined as S 0.G/DPfu;vgV.G/ d.u/d.v/R.u;v/, where d.u/ is the degree of vertex u, andR.u;v/ denotes the resistance distance between vertices u and v. In this paper we obtain someupper and lower bounds for the degree Kirchhoff index of graphs. We also obtain some boundsfor the Nordhaus-Gaddum-type result for the degree Kirchhoff index.2010 Mathematics Subject Classification: 05C12; 05C50Keywords: normalized Laplacian eigenvalues, degree Kirchhoff index, graphs1. INTRODUCTIONIn this paper all graphs considered are assumed to be connected. Let G D.V .G/;E.G// be a simple undirected graph with n D jV.G/j vertices and m DjE.G/j edges. For a vertex u2V.G/, d.u/ denote the degree of u. Dmaxfd.u/ju2V.G/g, ıDminfd.u/ju2V.G/g. The distance d.v;u/D d.u;vjG/ between the ver-tices v and u of the graph G is defined as the length of a shortest path between v andu. The diameter of G, denoted by D, is the maximum distance between all pairs ofvertices.The Wiener index is defined as the sum of distances between all unordered pairsof verticesW.G/DXfu;vgV.G/d.u;v/:This molecular structure descriptor is one of the most used topological indices, wellcorrelated with many physical and chemical properties of a variety of classes ofchemical compounds. For details, see the survey paper [9].This paper is supported by NNSFC (11101245, 11271208, 61202362, 11301302) and NSFSD(BS2013SF009, BS2010SF017, ZR2011AQ005).c 2014 Miskolc University Press98 LIHUA FENG, GUIHAI YU, AND WEIJUN LIUIn [16], a weighted version of the Wiener index is established, called the Gutmanindex [22, 23], which is defined asS.G/DXfu;vgV.G/d.u/d.v/d.u;v/ :For tree [16] it holds that S.G/D 4W.G/  .2n 1/.n 1/:The Gutman index of graphs attracted attention only quite recently. Several boundson the Gutman index are established in [1]. In [6] an asymptotic upper bound forS.G/ was reported. In [6] relations between the edge–Wiener index and Gutmanindex were established. The maximal and minimal Gutman indices of unicyclic andbicyclic graphs are determined in [4, 10] and [3, 12], respectively.In 1993 Klein and Randic´ [19] introduced a new distance function named resist-ance distance, based on the theory of electrical networks. They viewed G as an elec-trical network N by replacing each edge of G with a unit resistor. The resistance dis-tance between the vertices u and v of the graph G, denoted by R.u;v/DR.u;vjG/,is then defined to be the effective resistance between the nodes u and v in N . Similarto the long recognized shortest–path distance, the resistance distance is also intrinsicto the graph, not only with some nice purely mathematical properties, but also with asubstantial potential for chemical applications [18, 19, 25, 26].The Kirchhoff index (or resistance index) is defined in analogy to the Wiener indexas:Kf .G/DXfu;vgV.G/R.u;v/:The Kirchhoff index is also much studied in the literature. Extremal unicyclicgraphs with respect to the Kirchhoff index were determined in [15,27,28]. Deng alsostudied the Kirchhoff index of fully loaded unicyclic graphs [15] and graphs withmany cut edges [8]. Zhou [30] characterized the extremal graphs with given matchingnumber, connectivity, and minimal Kirchhoff index. Wang et al. [24] determined thefirst three minimal Kirchhoff indices among cacti.Recently, a new index named degree Kirchhoff index was put forward in [2]. It isdefined asS 0.G/DXfu;vgV.G/d.u/d.v/R.u;v/:Apparently, we can see that the degree Kirchhoff index may be viewed as theresistance–distance analogue of the Gutman index.Let LD .Luv/nn be the Laplacian matrix of the (connected) graph G. Its entryisLuv D8<: d.u/; if uD vI 1; if u v;0; otherwise:FURTHER RESULTS REGARDING THE DEGREE KIRCHHOFF INDEX OF GRAPHS 99Suppose the Laplacian eigenvalues of L.G/ are 1  2      n 1 >n D 0.For details on Laplacian eigenvalues see [13, 14, 21]. Then a long time known resultfor the Kirchhoff index is [17]:Kf .G/D nn 1XiD11i: (1.1)In [5] Fan Chung defined the normalized Laplacian matrixL .G/D .Luv/nn ofa graph G as follows:Luv D8<:1; if uD v;  1pd.u/d.v/; if u v;0; otherwise;We call eigenvalues of L .G/ the normalized Laplacian eiganvalues of G, and theeigenvalues of L .G/ are ordered by 1  2      n 1  n D 0. One can findthatL DD 1=2LD 1=2.A remarkable analogy between the Kirchhoff and degree Kirchhoff indices is theformula [2]:S 0.G/D 2mn 1XiD11i: (1.2)For the following consideration it is important to recall that R.u;v/ D R.v;u/,R.u;u/ D 0 and that [19] d.u;v/  R.u;v/ with equality if and only if there is aunique path linking the vertices u and v. Therefore, for trees, Kf .G/DW.G/ andS 0.G/D S.G/.In [31] the authors obtained the following result.Theorem 1. LetG be a connected (molecular) bipartite graph with n 2 verticesand m edges. ThenS 0.G/ .2n 3/mwith equality if and only if G ŠKr;n r for 1 r  bn2c.In [11] the upper and lower bounds the degree Kirchhoff index of unicyclic graphsis established. In this paper, we further study the degree Kirchhoff index of graphsand obtain several upper and lower bounds on the degree Kirchhoff index. We alsoobtain some bounds for the Nordhaus-Gaddum-type result for the degree Kirchhoffindex.2. LEMMAS AND RESULTSLemma 1 ([5]). Let G be a graph on n vertices. Then for n 2,1  nn 1;the equality holds if and only if G ŠKn.100 LIHUA FENG, GUIHAI YU, AND WEIJUN LIULemma 2 ([29]). Let G be a graph with the largest degree . Then1.G/ 1C 1;with equality if and only if G is a complete graph.Recall that the chromatic number .G/ of a graph G is the minimum number ofcolors such that G can be colored in a way such that no two adjacent vertices havethe same color.Lemma 3 ([5]). Let G be a graph on n vertices with chromatic number . Then1.G/  1:Lemma 4 ([7]). LetG be a connected graph of order n> 2. Then 2D   D n 1if and only if G ŠKn or G ŠKp;q .Theorem 2. Let G be a connected graph on n > 2 vertices and m edges. ThenS 0.G/ 2mn 2C 1n;S 0.G/ 2m C1C.n 2/2n 1  1!;S 0.G/ 2m C1C.n 2/2n 1  1!:Each of the above equalities holds if and only if G ŠKn.Proof. From the definition, one has12mS 0 Dn 1XiD11iD 11Cn 1XiD21i 11C .n 2/2Pn 1iD2 iD 11C .n 2/2n 1 :Now we consider the functionf .x/D 1xC .n 2/2n x :FURTHER RESULTS REGARDING THE DEGREE KIRCHHOFF INDEX OF GRAPHS 101It is easy to check that df .x/dx 0 for x  nn 1 , with equality if and only if xD nn 1 .Therefore it follows that f .x/ is not decreasing with respect to x. Since x  nn 1from Lemma 1, we havef .x/ f nn 1D n 2C 1n:This implies the result.If the equality holds, then all the inequalities in the proof must be equalities. Itfollows that 2 D   D n 1, and from Lemmas 4 and 1 we get the result. Thereverse is easy to check.Note that from the above proof and Lemmas 2, 3 we can getf .x/ fC1D C1C.n 2/2n 1  1;f .x/ f 1D C1C.n 2/2n 1  1:Therefore we obtain the desired two bounds involving the maximum degree or thechromatic number, which is larger than f nn 1. Fan Chung [5] obtained the following formula for the number of spanning trees ofa graph based on the eigenvalues of the normalized Laplacian:t .G/DQv d.v/Pv d.v/n 1YiD1i :Theorem 3. LetG be a connected graph with n 3 vertices, t spanning trees andmaximum vertex degree . ThenS 0.G/ 2m C1C .n 2/.C1/Qv d.v/2mt 1n 2!;with equality if and only if G ŠKn.Proof. From the definition, one has12mS 0 Dn 1XiD11iD 11Cn 1XiD21i 11C n 2.Qn 1iD2 i /1n 2102 LIHUA FENG, GUIHAI YU, AND WEIJUN LIUD 11C .n 2/ 1Qn 1iD1 i! 1n 2D 11C .n 2/1Qv d.v/2mt 1n 2:Letg.x/D 1xC .n 2/xQv d.v/2mt 1n 2:Then we havedg.x/dxD  1x2CxQv d.v/2mt 1n 2 1Qv d.v/2mtD  1x2C 1xxQv d.v/2mt 1n 2:Therefore g.x/ is non-decreasing forx 2mtQv d.v/ 1n 1:Note that2mtQv deg.v/ 1n 1 D n 1YiD1i! 1n 1Pn 1iD1 in 1 Dnn 1 C1 1;it follows thatg.x/ gC1D C1C .n 2/.C1/Qv d.v/2mt 1n 2:This implies the result. The equality case is easy to see from Lemma 4. One may note from the above proof that12mS 0 Dn 1XiD11i n 1.Qn 1iD1 i /1n 1D .n 1/Qv d.v/2mt 1n 1; (2.1)with equality if and only if 1 D 2 D   D n 1, that is G ŠKn.The above expression implies that S 0.G/ is strictly decreasing with respect tot .G/.Before listing an upper bound for t .G/, we first recall one definition. For k  1,a graph G is k-connected if either G is a complete graph KkC1, or G has at leastkC2 vertices and contains no .k 1/-vertex cut. The maximal value of k for whicha connected graph G is k-connected is the connectivity of G, denoted by .G/. If Gis disconnected, we define .G/D 0.If G is a graph of order n, then .G/  n 1, with equality holding if and only ifG ŠKn.FURTHER RESULTS REGARDING THE DEGREE KIRCHHOFF INDEX OF GRAPHS 103Recently, the following result for the spanning trees of graphs was obtained in[20].Let G be a connected graph of order n with connectivity .G/D k. Thent .G/ knk 1.n 1/n k 2:The equality holds if and only if G Š .K1[Kn k 1/_Kk :By using this bound and expression (2.1), we can get the following bound in-volving connectivity.Theorem 4. LetG be a connected graph with n 3 vertices. IfG has connectivity.G/D k, thenS 0  2m.n 1/ Qv d.v/2mknk 1.n 1/n k 2 1n 1;with equality if and only if G ŠKn.At last, we consider the Nordhaus-Gaddum-type result for the degree Kirchhoffindex.Theorem 5. Let G be a connected (molecular) graph on n  5 vertices with aconnected complement G. ThenS 0.G/CS 0.G/ .n 1/32:Proof. From the result in Theorem 2, we haveS 0.G/CS 0.G/ 2mn 2C 1nC  n2! 2m!n 2C 1nD n2!n 2C 1nD .n 1/32:This implies the result. Lemma 5 ([2]). Let G be a graph of order n, with m edges and diameter D. Forall vi ;vj 2 V.G/.i ¤ j /, thenR.vi ;vj / 2mD1d.vi /C 1d.vj /:Theorem 6. Let G be a connected (molecular) graph on n  5 vertices with dia-meter D. ThenS 0.G/ 4D.n 1/m2: (2.2)104 LIHUA FENG, GUIHAI YU, AND WEIJUN LIUProof. From the definition, we haveS 0.G/DXfu;vgV.G/d.u/d.v/R.u;v/Xfu;vgV.G/d.u/d.v/2mD1d.u/C 1d.v/D 2mDXfu;vgV.G/.d.u/Cd.v//D 2mD.n 1/Xu2V.G/d.u/D 4D.n 1/m2:This implies the result. Theorem 7. Let G be a graph of order n with maximum degree , and maximalGutman index S.G/. Then the following holds:S 0.G/ nC13!3: (2.3)Proof. Since d.u;v/  R.u;v/ with equality if and only if there is a unique pathlinking the vertices u and v [19], we immediately getS 0.G/DXfu;vgV.G/d.u/d.v/R.u;v/Xfu;vgV.G/d.u/d.v/d.u;v/D S.G/:In [1] it is obtained that under the hypothesis of the theorem,S.G/ nC13!3:The desired result is obtained. We now present one example to illustrate that the above two bounds are incompar-able. For the star graph H D K1;n 1, bound (2.2) reads 8.n  1/3, while bound (4)is .n  1/3 nC13, showing that (2.2) is better than (2.3). For the path Pn, bound (3)reads 4.n 1/4, while bound (2.3) is 8 nC13, showing that (2.3) is better than (2.2).Therefore the above two bounds are incomparable.In light of Theorems 6, 7, we can establish the following two upper bounds for theNordhaus-Gaddum-type results.FURTHER RESULTS REGARDING THE DEGREE KIRCHHOFF INDEX OF GRAPHS 105Theorem 8. Let G be a connected (molecular) graph on n  5 vertices with aconnected complement G. ThenS 0.G/CS 0.G/ 4.n 1/20@m2C  n2! m!21A :Proof. Note that the diameter of any graph is at most n  1. From the result inTheorem 6, we haveS 0.G/CS 0.G/ 4.n 1/20@m2C  n2! m!21A :This implies the result. Theorem 9. Let G be a connected (molecular) graph on n  5 vertices with aconnected complement G. ThenS 0.G/CS 0.G/ nC13! 3C .n 1  ı/3 :Proof. From the result in Theorem 7, we have the result. 3. FURTHER DISCUSSIONAs stated in Section 1, formulae (1.1) and (1.2) reveal that those two resistancedistance based molecular descriptors have a close connection with the eigenvaluesof some matrices associated with graphs. For a edge weighted connected graphG with weight cuv on the edge uv, and c.u/ DPuw2E.G/ cuw , we define S DPfu;vgV.G/ c.u/c.v/R.u;v/. Then we may naturally ask that whether there existsa real symmetric and positive semidefinite matrix P with eigenvalues1  2  : : : n 1 > n D 0such that S DPn 1iD1 1i ?Conversely, in general, letM be a real symmetric and positive semidefinite matrix.Suppose its eigenvalues are1  2  : : : n 1 > n D 0;with their corresponding mutually orthogonal unit eigenvector1; 2; : : : n 1; n:Now, we construct the matrixB DM C 1an.n/t ;106 LIHUA FENG, GUIHAI YU, AND WEIJUN LIUwhere aD .n/tn. We claim that the eigenvalues of B are1;2; : : : ;n 1;1:In fact, we can check that, for 1 i  n 1,Bi D .M C 1an.n/t /iDMi C 1an.n/tiDMiD ii :Similarly,Bn DMnC 1an.n/tn D 1an.n/tnD n:Since all the eigenvalues ofB are not zero,B is non-singular. Therefore, its inverseB 1 exists and its eigenvalues are 11; 12; : : : ; 1n 1 ;1:If we take M D LDD A, and n D 1, the all one vector, then we get Theorem1 and Theorem 2 in [26]. And the formulae (1) is established.If we take M DL DD 1=2LD 1=2, and n DD 12 1, then we get some part ofTheorem 3.1 in [2]. And the formulae (2) is established.Another natural question is: for general M described above, is there a connec-tion betweenPn 1iD1 1i and some (resistance) distance based molecular descriptor ofgraphs?We leave it for further research.REFERENCES[1] V. Andova, D. Dimitrov, J. Fink, and R. Sˇkrekovski, “Bounds on Gutman index,” MATCH Com-mun. Math. Comput. Chem., vol. 67, no. 2, pp. 515–524, 2012.[2] H. Chen and F. Zhang, “Resistance distance and the normalized Laplacian spectrum,” DiscreteAppl. Math., vol. 155, no. 5, pp. 654–661, 2007.[3] S. Chen and W. Liu, “Extremal modified Schultz index of bicyclic graphs,” MATCH Commun.Math. Comput. Chem., vol. 64, no. 3, pp. 767–782, 2010.[4] S. Chen and W. Liu, “Extremal university graphs with respect to modified Schultz index,” Util.Math., vol. 86, pp. 87–94, 2011.[5] F. R. K. Chung, Spectral graph theory, ser. 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Comput. Chem.,vol. 49, pp. 67–81, 2003.[26] W. Xiao and I. Gutman, “Resistance distance and Laplacian spectrum,” Theor. Chem. Acc., vol.110, pp. 284–289, 2003.[27] Y. Yang and X. Jiang, “Unicyclic graphs with extremal Kirchhoff index,” MATCH Commun. Math.Comput. Chem., vol. 60, no. 1, pp. 107–120, 2008.[28] W. Zhang and H. Deng, “The second maximal and minimal Kirchhoff indices of unicyclic graphs,”MATCH Commun. Math. Comput. Chem., vol. 61, no. 3, pp. 683–695, 2009.[29] X. Zhang, “On the Laplacian spectra of graphs,” Ars Comb., vol. 72, pp. 191–198, 2004.[30] B. Zhou and B. N. Trinajstic´, “The Kirchhoff index and the matching number,” Int. J. QuantumChem., vol. 109, pp. 2978–2981, 2009.[31] B. Zhou and N. Trinajstic´, “On resistance-distance and Kirchhoff index,” J. Math. Chem., vol. 46,no. 1, pp. 283–289, 2009.Authors’ addressesLihua FengCentral South University, Department of Mathematics, Changsha, Hunan, 410083, P.R. ChinaE-mail address: fenglh@163.com108 LIHUA FENG, GUIHAI YU, AND WEIJUN LIUGuihai YuShandong Institute of Business & Technology, School of Mathematics, Yantai, Shandong, P.R. ChinaE-mail address: yuguihai@126.comWeijun LiuCentral South University, Department of Mathematics, Changsha, Hunan, 410083, P.R. ChinaE-mail address: wjliu6210@163.com

Further results regarding the degree Kirchhoff index of graphs

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