Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

Graphs and Combinatorics (Impact Factor: 0.35). 01/2011; 27(2):215-229. DOI: 10.1007/s00373-010-0975-0
Source: DBLP

ABSTRACT In this paper, we consider the following problem. Over the class of all simple connected graphs of order n with k pendant vertices (n, k being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity
of unicyclic graphs.

KeywordsLaplacian matrix–Algebraic connectivity–Characteristic set–Perron component–Pendant vertex

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    ABSTRACT: In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.
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    ABSTRACT: Several approaches for ordering graphs by spectral parameters are presented in the literature. We can find graph orderings either by the greatest eigenvalue (spectral radius or index) or by the sum of the absolute values of the eigenvalues (the energy of a graph) or by the second smallest eigenvalue of the Laplacian matrix (the algebraic connectivity), among others. By considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs, several structural properties of graphs relative to this parameter have been studied. Hence, a large number of papers about ordering graphs by algebraic connectivity, mainly about trees and graphs with few cycles, have been published. This paper surveys the significant results concerning these topics, trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity.
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