Abstract
The Kirchhoff index is a new measure of network robustness. In this paper, we study the robustness of n × m mesh graphes (denoted by M n×m ) by determining the most important edges and the least important edges. In other words, we aim to find the edges (denoted by edge max ) which have the biggest impact on the Kirchhoff index after the edge is deleted and the edges (denoted by edge min ) which have the least impact on Kirchhoff index after the edge is deleted. The distributions of edge max and edge min are fully characterized. Consequently, we propose a new strategy called modified resistance distance strategy to locate edge max and edge min of M n×m . The applicability and rationality of the modified resistance distance strategy in mesh graphs is proved by comparing with other known strategies, such as the semi-random strategy, the degree product strategy and the resistance distance strategy. Moreover, the modified resistance distance strategy is still applicable in mesh graphs when we use the algebraic connectivity as the measure of graph robustness.
Highlights
Abundant entities can be abstracted into graphs, such as traffic network, warehouse storage network, YouTube social network and so on
We apply the modified resistance distance strategy to measure different mesh graphs robustness based on different graphs robustness measurements, such as the algebraic connectivity
We discover that the distribution of edgemax and edgemin when algebraic connectivity is used as graph robustness measure is the same as the distribution when Kirchhoff index is applied
Summary
Abundant entities can be abstracted into graphs, such as traffic network, warehouse storage network, YouTube social network and so on. Scholars propose graphs robustness measures based on different methods of calculating the robustness of graphs [1]: algebraic connectivity, Kirchhoff index, Wiener index, etc. We focus on selecting the edges have the most or least influence on graphs Kirchhoff index and we will compare those results with the one based on algebraic connectivity in order to verify the rationality and applicability of our strategy. We apply this new strategy to different mesh graphs on the purpose of validating its applicability. We apply the modified resistance distance strategy to measure different mesh graphs robustness based on different graphs robustness measurements, such as the algebraic connectivity. A conclusion addressing further research directions form the arguments of the last section
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