Author
Kamal Lochan Patra
Other affiliations: Homi Bhabha National Institute, Indian Institute of Technology Kanpur
Bio: Kamal Lochan Patra is an academic researcher from National Institute of Science Education and Research. The author has contributed to research in topics: Vertex (geometry) & Cyclic group. The author has an hindex of 7, co-authored 34 publications receiving 147 citations. Previous affiliations of Kamal Lochan Patra include Homi Bhabha National Institute & Indian Institute of Technology Kanpur.
Papers
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TL;DR: In this article, the Laplacian eigenvalues of the zero divisor graph Γ ( Z n ) of the ring Z n were studied and it was shown that Γ( Z n t ) is integral for every prime p and positive integer t ≥ 2.
32 citations
TL;DR: In this paper, it was shown that among all trees on n vertices, the path has the smallest algebraic connectivity and the star has the largest algebraic connection, and that the smallest and the largest paths have the largest algebras.
24 citations
16 Oct 2017
TL;DR: This paper is a survey on the upper and lower bounds for the largest eigenvalue of the LaPLacian matrix, known as the Laplacian spectral radius, of a graph, as functions of graph parameters.
Abstract: This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds are given as functions of graph parameters like the number of vertices, the number of edges, degree sequence, average 2-degrees, diameter, covering number, domination number, independence number and other parameters.
20 citations
TL;DR: A new upper bound is given for the vertex connectivity of $\kappa(\mathcal{P}(C_n)$ and it is determined when $n$ is a product of distinct prime numbers.
20 citations
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TL;DR: This paper considers the following problem: over the class of all simple connected graphs of order n with k pendant vertices, which graph maximizes or minimizes the algebraic connectivity?
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.
13 citations
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01 Jan 2012
TL;DR: A conjecture of AutoGraphiX on the relation between the Randic index R and the algebraic connectivity a of a connected graph G is: as mentioned in this paper, where R is defined as
Abstract: A conjecture of AutoGraphiX on the relation between the Randic index R and the algebraic connectivity a of a connected graph G is:
605 citations
01 Jan 2009
TL;DR: In this article, the authors introduce the concept of graph operations and modifications, and characterizations of spectra by characterizations by spectra and one eigenvalue, and Laplacians.
Abstract: Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.
398 citations
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TL;DR: In this paper, the adjacency matrix, a matrix of O's and l's, is used to store a graph or digraph in a computer, and certain matrix operations are seen to correspond to digraph concepts.
Abstract: In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O’s and l’s. The l’s correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.
292 citations
01 Jan 2009
151 citations
TL;DR: In this paper, the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices was studied in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges.
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 non-trivial 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.
60 citations