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Das, Kinkar Ch.,Mojallal, Seyed Ahmad Elsevier 2014 Discrete mathematics Vol.325 No.-
Let G be a graph with n vertices and m edges. Also let mu(1), mu(2), ..., mu(n-1), mu(n) = 0 be the eigenvalues of the Laplacian matrix of graph G. The Laplacian energy of the graph G is defined as LE = LE(G) = Sigma(n)(i=1) vertical bar mu(i) - 2m/n vertical bar. In this paper, we present some lower and upper bounds for LE of graph Gin terms of n, the number of edges m and the maximum degree Delta. Also we give a Nordhaus-Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs. (C) 2014 Elsevier B.V. All rights reserved.
On the Laplacian-energy-like invariant
Das, Kinkar Ch.,Gutman, Ivan,Ç,evik, A. Sinan Elsevier 2014 Linear Algebra and its Applications Vol.442 No.-
<P>Let G be a connected graph of order n with Laplacian eigenvalues mu(1) >= mu(2) >= ... mu(n-1) >mu(n) = 0. The Laplacian-energy-like invariant of the graph G is defined as LEL = LEL(G) = Sigma(n-1)(i=1)root mu(i) . Lower and upper bounds for LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees. (C) 2013 Elsevier Inc. All rights reserved.</P>
Extremal Laplacian energy of threshold graphs
Das, K.Ch.,Mojallal, S.A. Elsevier [etc.] 2016 Applied Mathematics and Computation Vol.273 No.-
<P>Let G be a connected threshold graph of order n with in edges and trace T. In this paper we give a lower bound on Laplacian energy in terms of n, in and T of G. From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies. (C) 2015 Elsevier Inc. All rights reserved.</P>
Characterization of graphs having extremal Randić indices
Das, Kinkar Ch.,Kwak, Jin Ho Elsevier 2007 Linear algebra and its applications Vol.420 No.1
<P><B>Abstract</B></P><P>The higher Randić index <I>R</I><SUB><I>t</I></SUB>(<I>G</I>) of a simple graph <I>G</I> is defined as<SUB>Rt</SUB>(G)=∑<SUB>i1</SUB><SUB>i2</SUB>⋯<SUB>it+1</SUB>1<SUB>δ<SUB>i1</SUB></SUB><SUB>δ<SUB>i2</SUB></SUB>⋯<SUB>δ<SUB>it+1</SUB></SUB>,where <I>δ</I><SUB><I>i</I></SUB> denotes the degree of the vertex <I>i</I> and <I>i</I><SUB>1</SUB><I>i</I><SUB>2</SUB>⋯<I>i</I><SUB>t+1</SUB> runs over all paths of length <I>t</I> in <I>G</I>. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339–344], the lower and upper bound on <I>R</I><SUB>1</SUB>(<I>G</I>) was determined in terms of a kind of Laplacian spectra, and the lower and upper bound on <I>R</I><SUB>2</SUB>(<I>G</I>) were done in terms of kinds of adjacency and Laplacian spectra. In this paper we characterize the graphs which achieve the upper or lower bounds of <I>R</I><SUB>1</SUB>(<I>G</I>) and <I>R</I><SUB>2</SUB>(<I>G</I>), respectively.</P>
Comparison between the zeroth-order Randic index and the sum-connectivity index
Das, K.Ch.,Dehmer, M. Elsevier [etc.] 2016 Applied Mathematics and Computation Vol.274 No.-
<P>The zeroth-order Randic index and the sum-connectivity index are very popular topological indices in mathematical chemistry. These two indices are based on vertex degrees of graphs and attracted a lot of attention in recent years. Recently Li and Li (2015) studied these two indices for trees of order n. In this paper we obtain a relation between the zeroth-order Randic index and the sum-connectivity index for graphs. From this we infer an upper bound for the sum-connectivity index of graphs. Moreover, we prove that the zeroth-order Randic index is greater than the sum-connectivity index for trees. Finally, we show that R-2,R-alpha(G) is greater or equal R-1,R-alpha(G) (alpha >= 1) for any graph and characterize the extremal graphs. (C) 2015 Elsevier Inc. All rights reserved.</P>
On energy and Laplacian energy of bipartite graphs
Das, K.Ch.,Mojallal, S.A.,Gutman, I. Elsevier [etc.] 2016 Applied Mathematics and Computation Vol.273 No.-
<P>Let G be a bipartite graph of order it with in edges. The energy epsilon(G) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for epsilon(G) in terms of 11, in, and detA. Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized. (C) 2015 Elsevier Inc. All rights reserved.</P>
On spectral radius and energy of extended adjacency matrix of graphs
Das, K.Ch.,Gutman, I.,Furtula, B. Elsevier [etc.] 2017 Applied Mathematics and Computation Vol.296 No.-
Let G be a graph of order n. For i=1,2,...,n, let d<SUB>i</SUB> be the degree of the vertex v<SUB>i</SUB> of G. The extended adjacency matrix A<SUB>ex</SUB> of G is defined so that its (i, j)-entry is equal to 12(d<SUB>i</SUB>d<SUB>j</SUB>+d<SUB>j</SUB>d<SUB>i</SUB>) if the vertices v<SUB>i</SUB> and v<SUB>j</SUB> are adjacent, and 0 otherwise,Yang et al. (1994). The spectral radius η<SUB>1</SUB> and the energy E<SUB>ex</SUB> of the A<SUB>ex</SUB>-matrix are examined. Lower and upper bounds on η<SUB>1</SUB> and E<SUB>ex</SUB> are obtained, and the respective extremal graphs characterized.
Relations between distance-based and degree-based topological indices
Das, K.Ch.,Gutman, I.,Nadjafi-Arani, M.J. Elsevier [etc.] 2015 Applied Mathematics and Computation Vol.270 No.-
Let W, Sz, PI, and WP be, respectively, the Wiener, Szeged, PI, and Wiener polarity indices of a molecular graph G. Let M<SUB>1</SUB> and M<SUB>2</SUB> be the first and second Zagreb indices of G. We obtain relations between these classical distance- and degree-based topological indices.