For a graph G, let ${\mu }_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that ${\mu }_2(G)+{\mu }_2(\bar{G})\ge 1$, where $\bar{G}$ is the complement of G. This conjecture has been proved for various families of graphs. Here, we prove this conjecture in the general case. Also, we will show that $\max \{{\mu }_2(G),{\mu }_2(\bar{G})\}\ge 1-O(n^{-\frac{1}{3}})$, where n is the number of vertices of G.

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关于图及其补的代数连通性和的下界

对于图G，让${\mu }_2(G)$表示其第二小的拉普拉斯特征值。据推测，${\mu }_2(G)+{\mu }_2(\bar{G})\ge 1$， 在哪里 $\bar{G}$是G的补码。这个猜想已经在各种图族中得到证明。在这里，我们在一般情况下证明这个猜想。此外，我们将证明$\mathrm{最大限度}\{{\mu }_2(G),{\mu }_2(\bar{G})\}\ge 1-哦(n^{-\frac{1}{3}})$，其中n是G的顶点数。