Some inequalities for the Kirchhoff index of graphs
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https://doi.org/10.26637/MJM0602/0008Abstract
Let $G$ be a simple connected graph of order $n$, sequence of vertex degrees $d_1 \geq d_2 \geq \cdots \geq d_n>0$ and Laplacian eigenvalues $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1}>\mu_n=0$. With $\Pi_1=\Pi_1(G)=\prod_{i=1}^n d_i^2$ we denote the multiplicative first Zagreb index of graph, and $K f(G)=n \sum_{i=1}^{n-1} \frac{1}{\mu_i}$ the Kirchhoff index of $G$. In this paper we determine several lower and upper bounds for $K f$ depending on some of the graph parameters such as number of vertices, maximum degree, minimum degree, and number of spanning trees or multiplicative Zagreb index.
Keywords:
Kirchhoff index, Laplacian eigenvalues (of graph), vertex degreeMathematics Subject Classification:
Mathematics- Pages: 349-353
- Date Published: 01-04-2018
- Vol. 6 No. 02 (2018): Malaya Journal of Matematik (MJM)
F. R. K. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, 1997.
I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total $pi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.
B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Zagreb indices: Bounds and Extremal graphs, In: Bounds in Chemical Graph Theory - Basics, (I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović, Eds.), Mathematical Chemistry Monographs, MCM 19, Univ. Kragujevac, Kragujevac, 2017, pp. 67-153.
B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17-100.
I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) $83-92$.
S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113-124.
D.W. Leea, S. Sedghib and N. Shobec, Zagreb Indices of a Graph and its Common Neighborhood Graph, Malaya J. Mat. 4(3) (2016) 468-475.
A. Ghalavand and A. R. Ashrafi, Extremal trees with respect to the first and second reformulated Zagreb index, Malaya J. Mat. 5 (3)(2017) 524-530.
R. Todeschini, D. Ballabio and V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, In: I. Gutman and B. Furtula (Eds.) Novel Molecular Structure Descriptors - Theory and Applications I (pp. 73-100), Mathematical Chemistry Monographs, MCM 8, Univ. Kragujevac, Kragujevac, 2010.
R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64(2) (2010) 359-372.
D. J. Klein and M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81-95.
I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996) 982-985.
H. Y. Zhu, D. J. Klein and I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996) $420-428$.
K. C. Das, On the Kirchhoff index of graphs, Z. Naturforsch 68a (2013) 531-538.
I. Gutman, B. Furtula, K. C. Das, E. Milovanović and I. Milovanović (Eds.), Bounds in Chemical Graph Theory - Basics, Mathematical Chemistry Monographs, MCM 19, Univ. Kragujevac, Kragujevac, 2017.
J. Liu, J. Cao, X. F. Pan and A. Elaiw, The Kirchhoff index of hypercubes and related complex networks, Discr: Dynam. Natur. Sci. (2013) Article ID 543189.
I. Milovanović, I. Gutman and E. Milovanović, On Kirchhoff and degree Kirchhoff indices, Filomat 29 (8) (2015) 1869-1877.
I. Ž. Milovanović and E. I. Milovanović, On some lower bounds of the Kirchhoff index, MATCH Commun. Math. Comput. Chem. 78 (2017) 169-180.
I. ̌̌. Milovanović and E. I. Milovanović, Bounds of Kirchhoff and degree Kirchhoff indices, In: Bounds in Chemical Graph Theory - Mainstreams (I. Gutman, B. Furtula, K. C. Das, E. Milovanović, I. Milovanović, Eds.), Mathematical Chemistry Monographs, MCM 20, Univ. Kragujevac, Kragujevac, 2017, pp. 93-119.
J. L. Palacios, Some additional bounds for the Kirchhoff index, MATCH Commun. Math. Comput. Chem. 75 (2016) 365-372.
B. Zhou and N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. $mathbf{4 5 5}$ (2008) 120-123.
M. Biernacki, H. Pidek and C. Ryll-Nardzewski, Sur une inegalite entre des integrales definies, Ann. Univ. Mariae Curie-Sklodowska A, 4 (1950), 1-4.
V. Cirtoaje, The best lower bound depended on two fixed variables for Jensen's inequality with ordered variables, J. Ineq. Appl., 2010 (2010) Article ID 128258, 1-12.
B. Zhou, I. Gutman and T. Aleksić, A note on the Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 441-446.
H. Kober, On the arithmetic and geometric means and on Hölder's inequality, Proc. Amer. Math. Soc. 9 (1958) $452-459$.
J. S. Li and Y. L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Lin. Multilin. Algebra 48 (2000) 117-121.
M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J, 37 (1987) 660-670.
R. Merris, Laplacian matrices of graphs: A survay, $mathrm{Lin}$. Algebra Appl., 197-198 (1994) 143-176.
K. C. Das, Sharp upper bound for the number of spanning trees of a graph, Graphs Comb. 23 (2007) 625-632.
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