Properties of first eigenvectors and eigenvalues of nonsingular weighted directed graphs
نویسندگان
چکیده
The class of nonsingular connected weighted directed graphs with an unweighted undirected branch is considered in this article. This paper investigates the monotonicity properties of the first eigenvectors of such graphs along certain paths. The paper describes how the first eigenvalue of such graphs changes under some perturbation. It is shown that replacing a branch which is a tree by a path on the same number of vertices will not increase the first eigenvalue, while replacing the tree by a star on the same number of vertices will not decrease the first eigenvalue. As an application the paper characterizes the graphs minimizing the first eigenvalue over certain classes of such graphs.
منابع مشابه
Lecture 12 : Introduction to Spectral Graph Theory , Cheeger ’ s inequality
Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Spectral graph theory seeks to relate the eigenvectors and eigenvalues of matrices corresponding to a Graph to the combinatorial properties of the graph. While generally the theorems are designed for unweighted and undirected graphs they can be extended to the weighted graph case and (less co...
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