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Circuits, Systems, and Signal Processing

03.12.2024

Extremal Values on the Kirchhoff Index of the Line Graph of Unicyclic Networks

verfasst von: Muhammad Shoaib Sardar, Shou-Jun Xu

Erschienen in: Circuits, Systems, and Signal Processing

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Abstract

Let G be a simple connected graph with vertex set V(G) and edge set E(G). Suppose N represents a network derived from G by substituting a 1-ohm resistor for each edge of G. In that case, the resistance between \(u, v \in V(G)\) is analogous to the resistance between two equivalent nodes in network N. The Kirchhoff index of G is the summation of the resistance distances between all pairs of vertices in G. The line graph \(L_G\) of G is a graph whose vertices correspond to the edges of G, and any two vertices of \(L_G\) are adjacent if and only if the corresponding edges of G are incident with the same vertex of G. A unicyclic graph is a connected graph containing exactly one cycle. In this paper, we will identify the extremal values and unicyclic graphs for the Kirchhoff index of the line graph of unicyclic graphs by utilizing techniques derived from electrical networks.

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Literatur
1.
W. Barrett, Emily. J. Evans, Amanda E. Francis, Resistance distance in straight linear 2-trees. Discrete Appl. Math. 258, 13-34 (2019)
2.
S.H. Bertz, The bond graph, J. Chem. Soc. Chem. Commun. 818-820 (1981)
3.
S.H. Bertz, A mathematical model of molecular complexity, in Chemical Applications of Topology and Graph Theory, (R B. King. editor), Elsevier, Amsterdam (1993)
4.
5.
B. Bollob\(\grave{a}\)s, Modern Graph Theory, New York, NY, USA: Springer-Verlag, 1998
6.
D. Bonchev, A.T. Balaban, X. Liu, D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50 (1), 1-20 (1994)
7.
H.Y. Chen, F.J. Zhang, Resistance distance and normalized Laplacian spectrum. Discrete Appl. Math. 155, 654–661 (2007)MathSciNetCrossRef
8.
P. Doyle, J. Snell, Random Walks and Electric Networks (American Mathematical Society, Providence, Rhode Island, 1984)CrossRef
9.
10.
11.
I. Gutman, J.H. Potgieter, Wiener index and intermolecular forces, J. Serb. Chem. Soc. 62 (1997)
12.
I. Gutman, Y.N. Yeh, S.L. Lee, Y.L. Luo, Some recent results in the theory of the Wiener number, lndian. J. Chem. 32A, 651–661 (1993)
13.
F. Harary, Graph theory (New Age International, New Delhi, 2001)
14.
F. Harary, Line Graphs, Graph Theory. Massachusetts: Addison-Wesley, 71-83 (1972)
15.
16.
G. Huang, W. He, Y. Tan, Theoretical and computational methods to minimize Kirchhoff index of graphs with a given edge \(k\)-partiteness. Appl. Math. Comput. 341, 348–357 (2019)MathSciNet
17.
A.E. Kennelly, Equivalence of triangles and stars in conducting networks. Electr. World Eng. 34, 413–414 (1899)
18.
19.
D.J. Klein, Resistance-distance sum rules. Croat. Chem. Acta 75, 633–649 (2002)
20.
D.J. Klein, Graph geometry, graph metrics and Wiener. MATCH COmmun. Math. Comput. Chem. 25, 7–27 (1997)MathSciNet
21.
22.
Q. Liu, The resistance distance and Kirchhoff index on quadrilateral graph and pentagonal graph. IEEE Access 7, 36617–36622 (2019)CrossRef
23.
J.B. Liu, X.F. Pan, Minimizing Kirchhoff index among graphs with a given vertex bipartiteness. Appl. Math. Comput. 291, 84–88 (2016)MathSciNet
24.
J.B. Liu, X.F. Pan, L. Yu, D. Li, Complete characterization of bicyclic graphs with minimal Kirchhoff index. Discret. Appl. Math. 200, 95–107 (2016)MathSciNetCrossRef
25.
Q. Liu, J.B. Liu, J. Cao, Further results on resistance distance and Kirchhoff index in electric networks. Discret. Dyn. Nat. Soc. 1, 4682527 (2016)MathSciNet
26.
I. Lukovits, S. Nikolic, N. Trinajstic, Resistance distance in regular graphs. Int. J. Quantum Chem. 71(3), 217–225 (1999)CrossRef
27.
S. Li, T. Tian, Resistance between two nodes of a ring clique network. Cicuits Syst. Signal Process. 41, 1287–1298 (2022)CrossRef
28.
B.H. McRae, B.G. Dickson, T.H. Keitt, V.B. Shah, Using circuit theory to model connectivity in ecology, evolution, and conservation. Ecology 89(10), 2712–2724 (2008)CrossRef
29.
M.F. Nadeem, F. Ishfaq, A. Shabbir, On resistance distance and Kirchhoff index of cacti networks. J. Stat. Phys. 191, 83 (2024)MathSciNetCrossRef
30.
J.L. Palacios, Resistance distance in graphs and random walks. Int. J. Quantum Chem. 81, 29–33 (2001)CrossRef
31.
W. Sajjad, X. F. Pan et al. Computation of resistance distance with Kirchhoff index of body centered cubic structure. J. Math. Chem., 1-20 (2024)
32.
W. Sajjad, M.S. Sardar, X.F. Pan, Computation of resistance distance and Kirchhoff index of chain of triangular bipyramid hexahedron. Appl. Math. Comput. 461, 128313 (2024)MathSciNet
33.
M.S. Sardar, H.-B. Hua, X.-F. Pan, H. Raza, On the resistance diameter of hypercubes. Physica A 540, 123076 (2020)MathSciNetCrossRef
34.
M. Sardar, J.B. Liu, I. Siddique, M. Jaradat, A novel and efficient method for computing the resistance distance. IEEE Access 9, 107104–107110 (2021)CrossRef
35.
M.S. Sardar, X.F. Pan, S.A. Xu, Computation of resistance distance and Kirchhoff index of the two classes of silicate networks. Appl. Math. Comput. 381, 125283 (2020)MathSciNet
36.
M.S. Sardar, X.F. Pan, S.J. Xu, Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs. Appl. Math. Comput. 473, 128670 (2024)MathSciNet
37.
M.S. Sardar, S.J. Xu, X.F. Pan, Extremal values on the Kirchhoff index of the line graph of trees. Kuwait J. Sci. 52(1), 100327 (2025)MathSciNetCrossRef
38.
39.
V. Stojanovic, V. Filipovic, Adaptive input design for identification of output error model with constrained output. Circuits Syst. Signal Process 33, 97–113 (2014)MathSciNetCrossRef
40.
V. Stojanovic, N. Nedic, D. Prsic, L. Dubonjic, Optimal experiment design for identification of ARX models with constrained output in non-Gaussian noise. Appl. Math. Model. 40(13–14), 6676–6689 (2016)MathSciNetCrossRef
41.
W. Sun, Y. Yang, A note on resistance distances of graphs. Front. Phy. 10, 896886 (2022)CrossRef
42.
W. Sun, Y. Yang, Extremal pentagonal chains with respect to the Kirchhoff index. Appl. Math. Comput. 437, 127534 (2023)MathSciNet
43.
M. Van Lier, R. Otten, Planarization by transformation. IEEE Trans. Circuit Theor. 20(2), 169–171 (1973)CrossRef
44.
K. Wang, W. Ning, On the degree resistance distance of unicyclic graphs. J. Appl. Math. Phys. 12, 3679–3689 (2024)CrossRef
45.
H. Wiener, Structural determination of paraffin boiling points. J. Amer. Chem. Soc. 69, 17–20 (1947)CrossRef
46.
W. Xiao, I. Gutman, Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110(4), 284–289 (2003)CrossRef
47.
S.A. Xu, Y.X. Li, H. Hua, X.F. Pan, On the resistance diameters of graphs and their line graphs. Discrete Appl. Math. 306, 174–185 (2022)MathSciNetCrossRef
48.
49.
Y. Yang, Y. Cao, H. Yao, J. Li, Solution to a conjecture on a Nordhaus-Gaddum type result for the Kirchhoff index. Appl. Math. Comput. 332, 241–249 (2018)MathSciNet
50.
Y. Yang, D.J. Klein, Resistance distance-based graph invariants of subdivisions and triangulations of graphs. Discrete Appl. Math. 181, 260–274 (2015)MathSciNetCrossRef
51.
Y. Yang, D.J. Klein, Comparison theorems on resistance distance and Kirchhoff index of \(S\), \(T\)-isomers. Discrete Appl. Math. 175, 87–93 (2014)MathSciNetCrossRef
52.
Y. Yang, X. Jiang, Unicyclic graphs with extremal Kirchhoff index. MATCH COmmun. Math. Comput. Chem. 60, 107–120 (2008)MathSciNet
53.
Y. Yang, V.R. Rosenfeld, Some spectral invariants of the neighbourhood corona of graphs. Discrete Appl. Math. 247, 300–308 (2018)MathSciNetCrossRef
54.
T. Zhang, C. Bu, Detecting community structure in complex networks via resistance distance. Physica A 526, 120782 (2019)CrossRef
55.
H.P. Zhang, Y.J. Yang, Resistance distance and Kirchhoff index in circulant graphs. Int. J. Quantum Chem. 107, 330–339 (2007)CrossRef
56.
Z. Zhu, J.B. Liu, The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes. Discrete Appl. Math. 254, 256–267 (2019)MathSciNetCrossRef
Metadaten
Titel
Extremal Values on the Kirchhoff Index of the Line Graph of Unicyclic Networks
verfasst von
Muhammad Shoaib Sardar
Shou-Jun Xu
Publikationsdatum
03.12.2024
Verlag
Springer US
Erschienen in
Circuits, Systems, and Signal Processing
Print ISSN: 0278-081X
Elektronische ISSN: 1531-5878
DOI
https://doi.org/10.1007/s00034-024-02924-7