IN1993, Klein and Randić [1] introduced a distance function

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1 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 Some esults of esistance Distance irchhoff Index Based on -Graph Qun Liu Abstract he resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor he irchhoff index fg is the sum of resistance distances between all pairs of vertices in G For a graph G let G be the graph obtained from G by adding a new vertex corresponding to each edge of G by joining each new vertex to the end vertices of the corresponding edge Let G G G G be the - vertex corona -edge corona of G G In this paper formulate for the resistance distance the irchhoff index in G G G G whenever G G are arbitrary graphs are obtained his improves extends some earlier results Index erms irchhoff index esistance distance -vertex corona -edge corona Generalized inverse I INODUCION IN993 lein ić [] introduced a distance function named resistance distance on the basis of electrical network theory he resistance distance r ij G between any two vertices i j in G is defined to be the effective resistance between them when unit resistors are placed on every edge of G he irchhoff index fg is the sum of resistance distances between all pairs of vertices of G he computation of two-point resistances in networks the irchhoff index are classical problem in electric theory graph theory he resistance distance the irchhoff index attracted extensive attention due to its wide applications in physics chemistry others For more information on resistance distance irchhoff index of graphs the readers are referred to efs []-[] the references therein Let G V G EG be a graph with vertex set V G edge set EG Let d i be the degree of vertex i in G D G diagd d d V G the diagonal matrix with all vertex degrees of G as its diagonal entries For a graph G let A G B G denote the adjacency matrix vertexedge incidence matrix of G respectively he matrix L G D G A G is called the Laplacian matrix of G where D G is the diagonal matrix of vertex degrees of G We use µ G u G µ n G 0 to denote the spectrum of L G If G is connected then any principal submatrix of L G is nonsingular In [3] new graph operations based on G graphs: - vertex corona -edge corona are introduced their A-spectrumresp L-spectrum are investigated For a graph G let G be the graph obtained from G by adding a new Manuscript received August 05; revised November 4 05 his work was supported by the National Natural Science Foundation of China Nos the Youth Foundation of exi University in Gansu ProvinceNo QN03-07 the eaching eform Project of exi University No XXXJY-04-0 Qun Liu is with the School of Mathematics Statistics exi University Zhangye Gansu China liuqun09@yeahnet vertex corresponding to each edge of G by joining each new vertex to the end vertices of the corresponding edge Let IG be the set of newly added vertices ie IG V G \ V G Let G G be two vertex-disjoint graphs Definition [3] he -vertex corona of G G denoted by G G is the graph obtained from vertex disjoint G V G copies of G by joining the ith vertex of V G to every vertex in the ith copy of G Definition [3] he -edge corona of G G denoted by G G is the graph obtained from vertex disjoint G IG copies of G by joining the ith vertex of IG to every vertex in the ith copy of G Note that if G i has n i vertices m i edges for i then G G has n + m + n n vertices 3m + n m + n n edges G G has n + m + m n vertices 3m + m m + m n edges his paper is organized as follows In Section some auxiliary lemma are given In Section 3 we obtain formulas for resistance distances of -vertex corona -edge corona of two arbitrary graphs In Section 4 we obtain formulas for irchhoff index of -vertex corona -edge corona of two arbitrary graphs II PELIMINAIES Let M be a square matrix M he {}-inverse of M is a matrix X such that MXM M If M is singular then M has infinitely many {}-inverse [4] he group inverse of M denoted by M is the unique matrix X such that MXM M XMX X MX XM It is known [9 ] that M exists if only if rankm rankm If M is real symmetric then M exists M is a symmetric {}- inverse of M Actually M is equal to the Moore-Penrose inverse of M since M is symmetric [5] We use M to denote any {}- inverse of a matrix M Let M uv -denote the u v-entry of M Lemma [] Let G be a connected graph hen r uv G L G uu + L G vv L G uv L G vu L G uu + L G vv L G uv Let n denotes the column vector of dimension n with all the entries equal one We will often use to denote an all-ones column vector if the dimension can be read from the context Lemma [7] For any graph we have L G 0 Lemma 3 [8] Let A B M C D Advance online publication: August 0

2 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 be a nonsingular matrix If A D are nonsingular then M A + A BS CA A BS S CA S A BD C A BS S CA S where S D CA B For a vertex i of a graph G let i denote the set of all neighbors of i in G Lemma 4 [7] i j V G r ij G d i + Lemma 5 [] vertices hen Let G be a connected graph For any k i r kj G d i kl i r kl G Let G be a connected graph on n fg ntrl G L G ntrl G Lemma [4] Let G be a connected graph of order n with edge set E hen Lemma 7 Let u<vuv E r uv G n A B L B D be the Laplacian matrix of a connected graph If D is nonsingular then X D B BD D + D B BD is a symmetric {}-inverse of L where ABD B Proof Let A BD B X BD D B D + D B BD I 0 0 D B I 0 D I BD Since D is nonsingular then L A B B D I BD By computation we have 0 0 D I 0 D B I LX I BD 0 0 D D B I I 0 0 D B I 0 D I BD I BD 0 I BD BD + BD LXL BD BD 0 I A B B D A BD B + BD B B B D L ence X is a symmetric {}-inverse of L where A BD B emarks: he above result is similar to Lemma 8 in [] this is another form of Lemma 8 but in the process of computing the resistance distance irchhoff index of graph G G G G we use this formula to be more superior than Lemma 8 in [] III ESISANCE DISANCE IN -VEEX COONA AND -EDGE COONA OF WO GAPS We first give formulae for resistance distance between two arbitrary vertexes in G G heorem Let G be a graph with n vertices m edges G be a graph with n vertices m edges hen the following holds: a For any i j V G we have r ij G G 3 L G ii + 3 L G jj 4 3 L G ij b For any i j V G we have r ij G G I n L G + I n ii + I n L G +I n jj I n L G + I n c For any i V G j V G we have r ij G G 3 L G ii + I n L G + I n jj 3 L G ij d For any i IG j V G V G let u i v i EG denote the edge corresponding to i we have r ij G G + r u i jg G + r v i jg G 4 r u i v i G G e For any i j IG let u i v i u j v j EG denote the edges corresponding to i j we have Advance online publication: August 0

3 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 r ij G G + 4 r u iu j G G + r uiv j G G +r vi u j G G + r vi v j G G r uiv i G G r ujv j G G Proof Let be the incidence matrix of G hen with a proper labeling of vertices the Laplacian matrix of G G G can be written as LG L G + D G + n I n I n n I m 0 m n n I n n 0 n n m I n L G + I n where 0 st denotes the s t matrix with all entries equal to zero Let A L G +D G +n I n B I n n B I n n I D m 0 m n n 0 n n m I n L G + I n We begin with the calculation about Let Q I n L G + I n then L G + D G + n I n I n n I m 0 m n n 0 n n m Q I n n L G + D G + n I n I n n L G + I n I n n L G + D G + n I n n I n 3 L G so we have 3 L G D B BD I m 0 m n n 0 nn m I n L G + I n I m 0 m n n 0 nn m I n L G + I n 4 where 3 L G 3 j n n L G Based on Lemma 7 the following matrix N 3 L G 3 L G 3 L G I m + L G 3 L G 3 L G 3 L G 3 L G Q + 3 L G is a symmetric {}- inverse of LG G where Q I n L G + I n I n n Let G G G then For any i j V G by Lemma the Equation we have r ij G 3 L G ii + 3 L G jj 4 3 L G ij as stated in a For any i j V G by Lemma the Equation we have r ij G I n L G + I n ii + I n L G +I n jj I n L G + I n ij Now we are ready to calculate BD D B Let I n n then BD I n n I m 0 m n n 0 n n m I n L G + I n I n n D B I m 0 m n n 0 nn m I n L G + I n I n n I n n Next we are ready to compute the D B BD as stated in b For any i V G j V G by Lemma the Equation we have r ij G 3 L G ii + I n L G + I n jj as stated in c 3 L G ij For any i IG j V G V G let u i v i EG denote the edge corresponding to i by Lemma 4 we have r ij G + r u i jg G + r v i jg G as stated in d 4 r u i v i G G For any i j IG let u i v i u j v j EG denote the Advance online publication: August 0

4 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 edges corresponding to i j by Lemma 4 we have r ij G + r u ijg G + r v ijg G as stated in e 4 r u i v i G G + rui u 4 j G G + r ui v j G G +r viu j G G + r viv j G G r ui v i G G r uj v j G G he following result is proved in a way that is certainly similar in spirit to the proof of heorem but is a little more complicated in detail Next we will give the formulate for the resistance distance between two arbitrary vertexes in G G heorem Let G be a graph on n vertices m edges G be a graph on n vertices m edges let G G G hen the following holds: a For any i j V G we have r ij G 3 L G ii + 3 L G jj 4 3 L G ij b For any i j V G we have r ij G I m L G + I n n + j n n ii +I m L G + I n n + j n n jj I m L G + I n n + j n n ij c For any i V G j V G we have r ij G 3 L G ii + I m L G + I n n + j n n jj I m L G + I n n + j n n ij d For any i IG j V G V G let u i v i EG denote the edge corresponding to i we have r ij G n + + r ju i G G + r jvi G G + r jk G G n + k V G fg + r uikg G + k V G r ui v i G G + k V G r vi kg G e For any i j IG let u i v i u j v j EG denote the edges corresponding to i j we have r ij G + riuj G G + r ivj G G n + + r ik G G n + k V G fg + r ujkg G k V G +r uj v j G G + k V G r vj kg G Proof Let be the incidence matrix of G hen with a proper labeling of vertices the Laplacian matrix of G G can be written as LG L G + D G 0 n m n n + I m I m n 0 mn n I m n I m L G + I n where 0 st denotes the s t matrix with all entries equal to zero Let A L G + D G B 0 n m n B 0 m n n D n + I m I m n I m n I m L G + I n Note that D G + A G Let [n + I m I m n I m L G + I n I m n ] I m By Lemma 3 we have D I m I m n I m n F where F I m L G + I n n + j n n We begin with the calculation about Let I m L G +I n n + j n n I m n then L G + D G 0 n m n I m I m n I m n L G + D G L G + D G 3 L G So we have 3 L G 0 m n n 0 m n n Now we are ready to calculate the BD D B BD 0 n m n I m I m n I m n I m L G + I n n + j n n Advance online publication: August 0

5 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 D B I m I m n I m n I m L G + I n n + j n n 0 m n n Next we are ready to compute the D B BD D B BD 0n m n I m 4 I m n I m n Based on Lemma 3 Lemma 7 the following matrix N 3 L G 3 L G 3 L G Q + Q + Q + 3 L G Q + I m 3 L G is a symmetric {}- inverse of LG G where I m n I m L G + I n n + j n n Q L G For any i j V G by Lemma the Equation we have r ij G G 3 L G ii + 3 L G jj 4 3 L G ij as stated in a For any i j V G by Lemma the Equation we have r ij G G I m L G + I n n + j n n ii as stated in b +I m L G + I n n + j n n jj I m L G + I n n + j n n ij For any i V G j V G by Lemma the Equation we have r ij G G 3 L G ii + I m L G + I n as stated in c n + j n n jj I m L G + I n n + j n n ij For any i IG j V G V G let u i v i EG denote the edge corresponding to i by Lemma 4 we have r ij G as stated in d n + r ju i G G + r jvi G G + + r jk G G n + k V G fg + r uikg G + k V G r ui v i G G + k V G r vi kg G For any i j IG let u i v i u j v j EG denote the edges corresponding to i j respectively By Lemma 4 we have r ij G as stated in e + riuj G G + r ivj G G n + + r ik G G n + k V G fg + r ujkg G + k V G r uj v j G G + k V G r vj kg G IV ICOFF INDEX IN -VEEX COONA AND -EDGE COONA OF WO GAPS In this section we will give the formulate for the irchhoff index in G G G G whenever G G are arbitrary graphs heorem 3 Let G be a graph with n vertices m edges G be a graph with n vertices m edges hen fg G + n n + m + n n fg 3n l + 3 trd n G L G + n i µ i G + + 3m n + π L G π 3 L G 3 L G n n m + n n where π d d d n I n n Proof Let L L G G hen G G be the symmetric {}-inverse of Advance online publication: August 0

6 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 trl G G 3 trl G + tr I m + L G + tr I n L G + I n + 3 tr L G 3 trl G + m + [L G ii i<jij EG +L G jj + L G ij ] + n trl G +I n + 3 trj n n L G Note that the eigenvalues of L G + I n are µ G + µ n G + then trl G + I n n i µ i G + By Lemma Lemma 5 we have trl G G 3n fg + m + i<jij EG [L G ii + L G jj r ij G ] + n n µ i G + + n fg 3n i + n 3n fg + m + 3 trd G L G n n + n µ i G + i Next we calculate the L G G Since L G 0 then L G G I m + L G + 3 L G + 3 L G + I n L G + I n + 3 L G Since π where π d d d n then L G π L G π Let n n I n L G + I n n n M L G + I n I n n then n n n M M M n n L G + I n n n n L G n n n n n I n n L G I n n nn n n n n L G n n n n n n n n L G n 0 so L G G m + π L G π + 3 L G Lemma 5 implies that fl G G + 3 L G + n n n + m + n n trl G G L G G hen plugging trl G G L G G into the e- quation above we obtain the required result Corollary Let G be an r - regular graph with n vertices m edges G be a graph with n vertices m edges hen + n + r fg G n + m + n n n fg + n i + 3m n + 3n µ i G + m + n n Proof Since r L G 0 then L G L G 0 π L G π r L G 0 hen the required result is obtained by plugging L G L G π L G π into heorem 3 heorem 4 Let G be a graph with n vertices m edges G be a graph with n vertices m edges hen fg G n + m + m n 3n fg + 3 trd G L G + tr L G n + m µ i G + + 3m n + i + 3n π L G π m + 4m n where I m n π d d d n Proof he proof is similar to that of heorem 3 hence we omit details Corollary Let G be an r -regular graph with n vertices m edges G be a graph with n vertices m edges hen + r fg G n + m + m n fg 3n + tr L G + m n i µ i G + + 3m n + 3 m + 4m n Advance online publication: August 0

7 IAENG International Journal of Applied Mathematics 4:3 IJAM_4_3_0 where I m n emark: From heorem 3 heorem 4 in [3] one can immediately get the irchhoff index of a graph by computing the Laplacian eigenvalues of the graph owever the expression given in heorem 3 heorem 4 in [3] are somewhat complicated In the above by the result of heorem 3 heorem 4 in a different way we obtain a much simpler expression V CONCLUSION We use the Laplacian generalized inverse approach to obtain the formulate for the resistance distance the irchhoff index in G G G G whenever G G are arbitrary graphs his approach is more simpler than the computation of the results in [3] improves extends some earlier results ACNOWLEDGMEN he author is very grateful to the anonymous referees for their carefully reading the paper for constructive comments suggestions which have improved this paper EFEENCES [] DJ lein M ić esistance distance J Math Chem vol pp [] B Bapat S Gupta esistance distance in wheels fans Indian J Pure Appl Math vol 4 no pp [3] Y Chen FJ Zhang esistance distance the normalized Laplacian spectrum Discrete Appl Math vol 55 pp [4] DJ lein esistance-distance sum rules Croat Chem Acta vol 75 pp [5] WJ Xiao I Gutman esistance distance Laplacian spectrum heor Chem Acc vol 0 pp [] YJ Yang DJ lein A recursion formula for resistance distances its applications Discrete Appl Math vol pp [7] P Zhang YJ Yang esistance distance irchhoff index in circulant graphs Int J Quantum Chem vol 07 pp [8] B Zhou N rinajstić On resistance-distance irchhoff index J Math Chem vol 4 pp [9] YJ Yang DJ lein esistance distance-based graph invariants of subdivisions triangulations of graphs Discrete Appl Math vol 8 pp [0] XG Liu J Zhou CJ Bu esistance distance irchhoff index of -vertex join -edge join of two Graphs Discrete Appl Math vol 87 pp [] W Gao L Shi Szeged related indices of unilateral polyomino chain unilateral hexagonal chain IAENG International Journal of Applied Mathematics vol 45 no pp [] J Liu X D Zhang Cube-connected complete graphs IAENG International Journal of Applied Mathematics vol 44 no 3 pp [3] J Lan B Zhou Spectra of graph operations based on -graph Linear Multilinear Algebra vol 3 no 7 pp [4] A Ben-Israel NE Greville Generalized Inverses: heory Applications nd ed Springer New York 003 [5] C Bu L Sun J Zhou YM Wei A note on block representations of the group inverse of Laplacian matrices Electron J Linear Algebra vol 3 pp [] LZ Sun WZ Wang J Zhou CJ Bu Some results on resistance distances resistance matrices Linear Multilinear Algebra vol 3 no 3 pp [7] CJ Bu B Yan XQ Zhou J Zhou esistance distance in subdivision-vertex join subdivision-edge join of graphs Linear Algebra Appl vol 458 pp [8] J Zhou LZ Sun WZ Wang CJ Bu Line star sets for Laplacian eigenvalues Linear Algebra Appl vol 440 pp Advance online publication: August 0