Journal of Applied Mathematics

Bounds for the Kirchhoff Index of Bipartite Graphs

Yujun Yang

Full-text: Open access

Abstract

A ( m , n ) -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D ( n , a , b ) consists of the path P n a b together with a independent vertices adjacent to one pendent vertex of P n a b and b independent vertices adjacent to the other pendent vertex of P n a b . In this paper, firstly, we show that, among ( m , n ) -bipartite graphs ( m n ) , the complete bipartite graph K m , n has minimal Kirchhoff index and the tree dumbbell D ( m + n , n (m + 1) / 2 , n (m + 1) / 2 ) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order l , the complete bipartite graph K {\lfloor} l / 2 {\rfloor} , l {\lfloor} l / 2 {\rfloor} has minimal Kirchhoff index and the path P l has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of ( m , n ) -bipartite graphs and bipartite graphs of order l are obtained by computing the Kirchhoff index of these extremal graphs.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 195242, 9 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495113

Digital Object Identifier
doi:10.1155/2012/195242

Mathematical Reviews number (MathSciNet)
MR2915714

Zentralblatt MATH identifier
1245.05107

Citation

Yang, Yujun. Bounds for the Kirchhoff Index of Bipartite Graphs. J. Appl. Math. 2012 (2012), Article ID 195242, 9 pages. doi:10.1155/2012/195242. https://projecteuclid.org/euclid.jam/1355495113


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