A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from {+/- 1, +/- i}

作者：Kalita Debajit; Pati Sukanta^*

来源：Linear Algebra and Its Applications, 2014, 449: 417-434.

DOI：10.1016/j.laa.2014.02.036

• Adjacency matrix
• Weighted directed graph
• Gain graph
• Property (SR) (strong reciprocal eigenvalue property)
• Property (SN) (strong negation eigenvalue property)
• Corona

A weighted directed graph is a directed graph G whose underlying undirected graph is simple and whose edges have nonzero (directional) complex weights, that is, the presence of an edge (u, v) of weight w is as good as the presence of the edge (v, u) with weight (w) over bar, the complex conjugate of w. Let G be a weighted directed graph on vertices 1, 2, ... , n. Denote by W-uv the weight of an edge (u, v) is an element of E(G). The adjacency matrix A(G) of G is an n x n matrix with entries a(ij) = w(ij) or (w) over bar (ji) or 0, depending on whether (i, j) is an element of E(G) or (j, i) is an element of E(G) or otherwise, respectively. We supply a characterization of those unicyclic weighted directed graphs G whose edges have weights from the set {+/- 1, +/- i} and whose adjacency matrix A(G) satisfies the following property: 'lambda is an eigenvalue of A(G) with multiplicity Is if and only if 1/lambda is an eigenvalue of A(G) with the same multiplicity'.

• 出版日期2014-5-15