Kirchhoff index in line, subdivision and total graphs of a regular graph

“…The resistance distance and the Kirchhoff index attracted extensive attention due to their wide applications in electric network theory, physics, chemistry, etc. and the Kirchoff index of graphs constructed by graph operations was also obtained; see [8,16,21,28,30,31,32].…”

Section: Introductionmentioning

confidence: 99%

Spectra of (H1,H2)-merged subdivision graph of a graph

Rajkumar

Gayathri

2019

Indagationes Mathematicae

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In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, R−graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi et al., 2017), and consequently, Q−graph, total graph, and quasitotal graph. We denote this new graph by [S(G)] H1 H2 , where G is a graph and, H 1 and H 2 are suitable graphs corresponding to G. Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of [S(G)] H1 H2 for some classes of graphs G, H 1 and H 2 . From these results, we derive the L-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.

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“…Its basic properties can be found in any text book on graph theory (see, e.g., [21][22][23]). Recently, several papers on line graph have been published [20,[24][25][26][27]. For example, Gao et al [24] established a formula and lower bounds for the Kirchhoff index of the line graph of a regular graph.…”

Section: Journal Of Applied Mathematicsmentioning

confidence: 99%

“…Recently, several papers on line graph have been published [20,[24][25][26][27]. For example, Gao et al [24] established a formula and lower bounds for the Kirchhoff index of the line graph of a regular graph. Bounds for Laplacian-energylike invariant (LEL for short) of the line graph of a regular graph are obtained in [26].…”

Section: Journal Of Applied Mathematicsmentioning

confidence: 99%