Alternative Proofs on the Indices of Cacti and Unicyclic Graphs with n Vertices

2011/10/12 by Sudipta Mallik, Mallik, Sudipta
Chemistry · Computer Science · Mathematics · #Graph Labeling and Dimension Problems #Graph theory and applications #Synthesis and Properties of Aromatic Compounds #math.CO

paper · pdf · doi:10.48550/arxiv.1110.2571

10 pages, 3 figures

arxiv created 2011/10/18 · arxiv updated 2011/10/19

Abstract

Let H_n be the cactus obtained from the star K_1,n-1 by adding \lfloor (n-1)/(2)\rfloor independent edges between pairs of pendant vertices. Let K_1,n-1⁺ be the unicyclic graph obtained from the star K_1,n-1 by appending one edge. In this paper we give alternative proofs of the following results: Among all cacti with n vertices, H_n is the unique cactus whose spectral radius is maximal, and among all unicyclic graphs with n vertices, K_1,n-1⁺ is the unique unicyclic graph whose spectral radius is maximal. We also prove that among all odd-cycle graphs with n vertices, H_n is the unique odd-cycle graph whose spectral radius is maximal.

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