The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm-based ergodicity coefficients $\left\{{\tau }_p\right\}$. If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using $\left\{{\tau }_p\right\}$. In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for ${\tau }_p$ and Theorem 4.7 says that ${\tau }_1$ is less than or equal to ${\tau }_{\mathrm{\infty }}$ for the Laplacian matrix of every simple graph. An application related to the stability of Markov chains is discussed. Other discussions, open questions and examples are provided.

主要结果是推论2.9，该推论使用基于矢量范数的遍历系数，为实常数行和矩阵的绝对值提供了更大的极限值，甚至更好，逼近了最大非平凡的特征值 $\left\{{\tau }_p\right\}$。如果常数行和矩阵不是奇异的，则还说明了如何使用来限制其绝对值中最小的非平凡特征值$\left\{{\tau }_p\right\}$。在最后一部分中，将这两个结果应用于绑定拉普拉斯矩阵的谱半径及其关联图的代数连通性。获得了许多其他结果。特别地，定理2.15是${\tau }_p$ 定理4.7说 ${\tau }_{\mathrm{1个}}$ 小于或等于 ${\tau }_{\mathrm{\infty }}$每个简单图的拉普拉斯矩阵。讨论了与马尔可夫链的稳定性有关的应用。提供了其他讨论，开放性问题和示例。