Article
Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices
Graphs and Combinatorics
(Impact Factor: 0.33).
03/2011;
27(2):215229.
DOI: 10.1007/s0037301009750
Source: arXiv
Fulltext
A. K. Lal, May 09, 2014 Available from:
Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed.
The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual
current impact factor.
Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence
agreement may be applicable.

[Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain nontrivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature. 
[Show abstract] [Hide abstract]
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.Linear Algebra and its Applications 10/2014; 458:429–453. DOI:10.1016/j.laa.2014.06.016 · 0.98 Impact Factor 
[Show abstract] [Hide abstract]
ABSTRACT: The distance Laplacian matrix of a connected graph G is defined in and and it is proved that for a graph G on n vertices, if the complement of G is connected, then the second smallest distance Laplacian eigenvalue is strictly greater than n . In this article, we consider the graphs whose complement is a tree or a unicyclic graph, and characterize the graphs among them having n+1n+1 as the second smallest distance Laplacian eigenvalue. We prove that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has the similar property like that of a Fiedler vector.Linear Algebra and its Applications 11/2014; 460:97–110. DOI:10.1016/j.laa.2014.07.025 · 0.98 Impact Factor