## Journal of Applied Mathematics

### Bounds for the Kirchhoff Index of Bipartite Graphs

Yujun Yang

#### 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 $(m\le n)$, the complete bipartite graph ${K}_{m,n}$ has minimal Kirchhoff index and the tree dumbbell $D(m+n,{\lfloor}n-\mathrm{(m}+1)/2{\rfloor},{\lceil}n-\mathrm{(m}+1)/2{\rceil})$ 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

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