| Summary: | The Kirchhoff index is a new measure of network robustness. In this paper, we study the robustness of <inline-formula> <tex-math notation="LaTeX">$n \times m$ </tex-math></inline-formula> mesh graphes (denoted by <inline-formula> <tex-math notation="LaTeX">$M_{n\times m}$ </tex-math></inline-formula>) by determining the most important edges and the least important edges. In other words, we aim to find the edges (denoted by <inline-formula> <tex-math notation="LaTeX">$edge_{max}$ </tex-math></inline-formula>) which have the biggest impact on the Kirchhoff index after the edge is deleted and the edges (denoted by <inline-formula> <tex-math notation="LaTeX">$edge_{min}$ </tex-math></inline-formula>) which have the least impact on Kirchhoff index after the edge is deleted. The distributions of <inline-formula> <tex-math notation="LaTeX">$edge_{max}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$edge_{min}$ </tex-math></inline-formula> are fully characterized. Consequently, we propose a new strategy called modified resistance distance strategy to locate <inline-formula> <tex-math notation="LaTeX">$edge_{max}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$edge_{min}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$M_{n\times m}$ </tex-math></inline-formula>. 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.
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