The Kirchhoff energy and Kirchhoff Laplacian energy for Kirchhoff matrix are examined in this paper. The Kirchhoff index with Kirchhoff Laplacian eigenvalues is defined and different inequalities including the distances, the vertices and the edges are obtained. Indeed, some bounds for the degree Kirchhoff index associated with its eigenvalues are found.
B. Acar, Introducing The New Kirchhoff Structures over Graphs, Msc Thesis, Selcuk University, Türkiye,
B. Zhou, I. Gutman and T. Aleksić, A note on the Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441–446.
B. Zhou and N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. 455 (1−3) (2008) 120–123.
B. Bozkurt and D. Bozkurt, On the sum of powers of normalized Laplacian eigenvalues of graphs, MATCH Commun.Math. Comput. Chem. 68 (3) (2012) 917–930.
D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs, Academic Press, New York,
F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 92. American. Mathematical Society, Providence, RI, 1997.
G. Yu and L. Feng, Randić index and eigenvalues of graphs, Rocky Mountain J. Math. 40 (2010) 713–721.
H. Chen and F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math. 155 (5) (2007) 654–661.
H. Kober, On the arithmetic and geometric means and on Hölder's inequality, Proc. Amer. Math. Soc. 9 (1958) 452–459.
H. Y. Zhu, D. J. Klein and I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) 420–428.
I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996) 982–985.
I. Gutman, B. Zhou and B. Furtula, The Laplacian-energy like invariant is an energy like invariant, MATCH Commun. Math. Comput. 64 (1) (2010) 85–96.
I. Gutman, The energy of a graph, Ber. –Statist. Sekt. Forsch. Graz 103 (1978) 1–22.
I. Gutman, The energy of a graph: old and new results, Algebraic combinatorics and applications (Gößweinstein, 1999), 196–211, Springer, Berlin,
I. Ž. Milovanović, E. I. Milovanović and E. Glogić, Lower bounds of the Kirchhoff and degree Kirchhoff index, Publ. State Univ. Novi Pazar, Ser. A, Appl. Math. Inf. Mech.7 (2015) 25-31.
I. Ž. Milovanović, E. I. Milovanović, E. Glogić and M. Matejić, On Kirchhoff index, Laplacian energy and their relations, MATCH Commun. Math. Comput. 81 (2019) 405–418.
I. Ž. Milovanović and E. I. Milovanović, On some lower bounds of the Kirchhoff index, MATCH Commun. Math. Comput. Chem. 78 (2017) 169-
J. Liu and B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. 59 (2008) 397–419.
K. C. Das, A sharp upper bound for the number of spanning trees of a graph, Graphs Combin. 23 (6) (2007) 625–632.
K. C. Das, On the Kirchhoff index of graphs, Naturforsch 68a (8−9) (2013) 531–538.
K. C. Das, A. D. Güngör and A. S. Çevik, On Kirchhoff index and resistance distance energy of a graph, MATCH Commun. Math. Comput. 67 (2) (2012) 541–556.
Gök, G. (2022). Kirchhoff Index and Kirchhoff Energy. Iranian Journal of Mathematical Chemistry, 13(3), 175-185. doi: 10.22052/ijmc.2022.246278.1619
MLA
Gülistan Kaya Gök. "Kirchhoff Index and Kirchhoff Energy". Iranian Journal of Mathematical Chemistry, 13, 3, 2022, 175-185. doi: 10.22052/ijmc.2022.246278.1619
HARVARD
Gök, G. (2022). 'Kirchhoff Index and Kirchhoff Energy', Iranian Journal of Mathematical Chemistry, 13(3), pp. 175-185. doi: 10.22052/ijmc.2022.246278.1619
VANCOUVER
Gök, G. Kirchhoff Index and Kirchhoff Energy. Iranian Journal of Mathematical Chemistry, 2022; 13(3): 175-185. doi: 10.22052/ijmc.2022.246278.1619
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