Irregular graph, eigenvector, principal ratio
The principal ratio of a connected graph, denoted γ(G), is the ratio of the maximum and minimum entries of its Perron eigenvector. Cioaba and Gregory (2007) conjectured that the graph on n vertices maximizing γ(G) is a kite graph, that is, a complete graph with a pendant path. In this paper, their conjecture is proved
Tait, Michael and Tobin, Josh.
"Characterizing Graphs Of Maximum Principal Ratio",
Electronic Journal of Linear Algebra,
Volume 34, pp. 61-70.