On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree

Ou, Jianping; Feng, Xiaoliang; Liu, Saihua

doi:10.1155/2014/316108

Journal of Applied Mathematics

2014

DOI: 10.1155/2014/316108

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On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree

Jianping Ou

Xiaoliang Feng

Saihua Liu

Abstract: The Wiener polarity index of a connected graph is defined as the number of its pairs of vertices that are at distance three. By introducing some graph transformations, in different way with that of Huang et al., 2013, we determine the minimum Wiener polarity index of unicyclic graphs with any given maximum degree and girth, and characterize extremal graphs. These observations lead to the determination of the minimum Wiener polarity index of unicyclic graphs and the characterization of the extremal graphs.

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2016

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Cited by 1 publication

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Saturation in the Hypercube and Bootstrap Percolation

Morrison¹,

Noel²,

Scott³

2016

Combinator. Probab. Comp.

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Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

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Saturation in the Hypercube and Bootstrap Percolation

Morrison¹,

Noel²,

Scott³

2016

Combinator. Probab. Comp.

View full text Add to dashboard Cite

show abstract

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On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree

Cited by 1 publication

References 7 publications

Saturation in the Hypercube and Bootstrap Percolation

Saturation in the Hypercube and Bootstrap Percolation

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