The degree resistance distance of cacti

Du, Junfeng; Su, Guifu; Tu, Jianhua; Gutman, Iván

doi:10.1016/j.dam.2015.02.022

Cited by 19 publications

(20 citation statements)

References 21 publications

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“…According to the above discussion, we find that the extremal cacti for the index R * (G) are the same as the extremal cacti for the Kirchhoff index, the multiplicative degree-Kirchhoff index, the Wiener index and the other indices [22,29,34,35]. Based on the known results for these indices, we guess the element of Cact(n; t) with maximum multiplicative degree-Kirchhoff index is isomorphic to the graph C n,t (as shown in Figure 6).…”

Section: Resultsmentioning

confidence: 65%

“…The indices have been well studied in both mathematical and chemical literature. In [22] some properties of R + (G) are determined and the extremal graph of cacti with minimum R + -value characterized. Bianchi et al [23] studied some upper and lower bounds for G + (G) whose expressions do not depend on the resistance distances.…”

Section: Introductionmentioning

confidence: 99%

“…Yang and Klein [26] derived a formula for R * (G) of subdivisions and triangulations of graphs. To simplify the calculation of R * (G), the present authors [27] also obtained a formula for R * (G) with respect to the subgraph of G. For more work on the topological indices, we refer the reader to [13,21,22,[28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

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The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Zhu

2019

Mathematics

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For a graph G, the resistance distance r G ( x , y ) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ ( G ) = ∑ { x , y } ⊂ V ( G ) d G ( x ) d G ( y ) r G ( x , y ) , where d G ( x ) is the degree of vertex x, and V ( G ) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t ( n ; t ) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ ( G ) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

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Section: Resultsmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

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The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Zhu

2019

Mathematics

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“…Later, the unicyclic graphs with the maximum and second-maximum D R -value were considered in [19,20]. In [21,22], the cactus graphs with the minimum, the second-minimum, and the thirdminimum D R -values were also completely characterized. Recently, the bicyclic graphs with maximum and minimum D R -values were determined in [23,24], respectively.…”

Section: Denote the Minimum Degree Of Vertices In G By δ(G)mentioning

confidence: 99%

Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

Ning

Wang

Raza

2021

Journal of Mathematics

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Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.

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“…The resistance distance between the vertices v i and v j , denoted by r(v i , v j ) (if more than one graphs are considered, we write r G (v i , v j ) to avoid confusion), is defined to be the effective resistance between the vertices v i and v j in G. If the ordinary distance is replaced by resistance distance in the expression for the Wiener index, one arrives at the Kirchhoff index [1,2] K f (G) = ∑ {u,v}⊆V(G)…”

Section: Introductionmentioning

confidence: 99%

Further Results on the Resistance-Harary Index of Unicyclic Graphs

Chen

Liu

et al. 2019

Mathematics

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The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.

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The degree resistance distance of cacti

Cited by 19 publications

References 21 publications

The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

Further Results on the Resistance-Harary Index of Unicyclic Graphs

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