Kirchhoff index of composite graphs

Zhang, Heping; Yang, Yujun; Li, Chuanwen

doi:10.1016/j.dam.2009.03.007

Cited by 92 publications

(57 citation statements)

References 25 publications

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“…Motivated by the results, in this paper we considered the generalization of the R-vertex corona and the R-edge corona to the case of n(m) different graphs and we obtain the resistances distance and the Kirchhoff index in terms of the corresponding parameters of the factors. These results generalize the existing results in [9].…”

Section: Introductionsupporting

confidence: 91%

Resistance distance and Kirchhoff index in generalized R-vertex and R-edge corona for graphs

Liu

2019

Filomat

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For a graph G, the graph R(G) of a graph G is the graph obtained by adding a new vertex for each edge of G and joining each new vertex to both end vertices of the corresponding edge. Let I(G) be the set of newly added vertices, i.e I(G) = V(R(G))\ V(G). The generalized R-vertex corona of G and Hi for i = 1, 2, ?,n, denoted by R(G) ?? ^n i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of V(G) to every vertex in Hi. The generalized R-edge corona of G and Hi for i = 1, 2, ?,m, denoted by R(G)?^m i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of I(G) to every vertex in Hi. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of R(G) ?? ^n i=1 Hi and R(G) ? ^m i=1 Hi whenever G and Hi are arbitrary graphs. These results generalize the existing results.

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Section: Introductionsupporting

confidence: 91%

Resistance distance and Kirchhoff index in generalized R-vertex and R-edge corona for graphs

Liu

2019

Filomat

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“…K f (G) = u<v r uv (G). Some results on resistance distance and Kirchhoff index can be found in [1][2][3][4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Some results on resistance distances and resistance matrices

Sun

Wang

Zhou

et al. 2014

Linear and Multilinear Algebra

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“…In Theorem 2.7, if x = y, then G = T (G 1 ) is the cluster T {G 1 } as defined in [9] and it can be easily checked that the formula for Kf(G) in the above theorem and in Theorem 3.11 of [9] coincide. Note that in [9] all resistances of edges are assumed to be unit.…”

Section: Definition 21mentioning

confidence: 82%

“…The concept of resistance distance was first introduced by Klein and Randić [4]. Recently this concept has got a wide attention from different authors especially those interested in applications in quantum chemistry, see for example [1][2][3]5,[7][8][9].…”

Section: Introductionmentioning

confidence: 99%

“…In §2, we define a composition of T and G (calling it the T -repetition of G) and compute its Kirchhoff index in terms of parameters of T and G. In §3, we study how Kf(G) behaves under some graph operations such as joining vertices and deleting or subdividing edges. In [9], the Kirchhoff index of some composite graphs, including the join of two arbitrary graphs is computed. But in contrast to that paper, here all ρ e 's are assumed to be units.…”

Section: Introductionmentioning

confidence: 99%

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