We study the behavior of <italic>algebraic connectivity</italic> in a weighted graph that is subject to <italic>site percolation</italic>, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for <inline-formula><tex-math notation="LaTeX">$(n,d,\lambda)$</tex-math><alternatives> <inline-graphic xlink:href="bahmani-ieq1-2757762.gif"/></alternatives></inline-formula>-graphs (i.e., <inline-formula> <tex-math notation="LaTeX">$d$</tex-math><alternatives><inline-graphic xlink:href="bahmani-ieq2-2757762.gif"/> </alternatives></inline-formula>-regular graphs on <inline-formula><tex-math notation="LaTeX">$n$</tex-math> <alternatives><inline-graphic xlink:href="bahmani-ieq3-2757762.gif"/></alternatives></inline-formula> vertices with all non-trivial eigenvalues of the adjacency matrix less than <inline-formula><tex-math notation="LaTeX">$\lambda$ </tex-math><alternatives><inline-graphic xlink:href="bahmani-ieq4-2757762.gif"/></alternatives></inline-formula> in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability <inline-formula><tex-math notation="LaTeX">$p\geq1-C_{1}n^{-C_{2}/d}$</tex-math> <alternatives><inline-graphic xlink:href="bahmani-ieq5-2757762.gif"/></alternatives></inline-formula> with <inline-formula><tex-math notation="LaTeX">$C_{1}$</tex-math><alternatives> <inline-graphic xlink:href="bahmani-ieq6-2757762.gif"/></alternatives></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$C_{2}$</tex-math><alternatives><inline-graphic xlink:href="bahmani-ieq7-2757762.gif"/> </alternatives></inline-formula> depending only on <inline-formula><tex-math notation="LaTeX">$\lambda /d$</tex-math> <alternatives><inline-graphic xlink:href="bahmani-ieq8-2757762.gif"/></alternatives></inline-formula>.
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