Independence number and spectral radius of cactus graphs

Abstract

Four Color Theorem implies that the independence number of every planar graph is at least n/4. A cactus graph is a connected planar graph whose block is either an edge or a cycle. In this paper, the lower bound for independence number of cactus graphs is improved to ⌈n/3⌉. Moreover, the maximum spectral radius of cactus graphs with fixed independence number is determined.