Kinkar Ch. Das

An improved upper bound for Laplacian graph eigenvalues

Abstract Let G=(V,E) be a simple graph on vertex set V={v1,v2,…,vn}. Further let di be the degree of vi and Ni be the set of neighbors of vi. It is shown that max d i +d j −|N i ∩N j |:1⩽i v i v j ∈E is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where |Ni∩Nj| denotes the number of common neighbors between vi and vj. For any G, this bound does not exceed the order of G. Further using the concept of common neighbors another upper bound for the largest eigenvalue of the Laplacian matrix of a graph has been obtained as max 2 d 2 i +d i m ′ i :1⩽i⩽n , where m ′ i = ∑ j d j −|N i ∩N j |:v i v j ∈E d i .

The largest two Laplacian eigenvalues of a graph

In this article, we present lower bounds for the largest eigenvalue, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph.In this article, we present lower bounds for the largest eigenvalue, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph.

Maximum eigenvalue of the reciprocal distance matrix

In this paper, we obtain the lower and upper bounds of the maximum eigenvalue of the reciprocal distance matrix of a connected (molecular) graph. We also give the Nordhaus-Gaddum-type result for the maximum eigenvalue.

The Harary Index of a graph

Introduction.- Extremal Graphs with Respect to Harary Index.- Relation Between the Harary Index and Related Topological Indices.- Some Properties and Applications of Harary Index.- The Variants of Harary Index.- Open Problems.

Weighted Harary indices of apex trees and k -apex trees

If G is a connected graph, then H A ( G ) = ? u ? v ( deg ( u ) + deg ( v ) ) / d ( u , v ) is the additively Harary index and H M ( G ) = ? u ? v deg ( u ) deg ( v ) / d ( u , v ) the multiplicatively Harary index of G . G is an apex tree if it contains a vertex x such that G - x is a tree and is a k -apex tree if k is the smallest integer for which there exists a k -set X ? V ( G ) such that G - X is a tree. Upper and lower bounds on H A and H M are determined for apex trees and k -apex trees. The corresponding extremal graphs are also characterized in all the cases except for the minimum k -apex trees, k ? 3 . In particular, if k ? 2 and n ? 6 , then H A ( G ) ? ( k + 1 ) ( 3 n 2 - 5 n - k 2 - k + 2 ) / 2 holds for any k -apex tree G , equality holding if and only if G is the join of K k and K 1 , n - k - 1 .

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Abstract For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants and named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

Improved upper and lower bounds for the spectral radius of digraphs

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius @r(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on @r(G) are given. We show that some known bounds can be obtained from our bounds.

Orthogonality and the numerical range. II

Abstract For a continuous linear operator A on a Hilbert space X and unit vectors x and y, an investigation of the set W[x,y]={z ∗ Az:z ∗ z=1 and zϵspan{x,y}} reveals several new results about W(A), the numerical range of A. W[x,y] is an elliptical disk (possibly degenerate), and several conditions are given which imply that W[x,y] is a line segment. In particular if x is a reducing eigenvector of A, then W[x,y] is a line segment. A unit vector is called interior (boundary) if x∗Ax is in the interior (boundary) of W(A). It is shown that interior reducing eigenvectorsare orthogonal to all boundary vectors and that boundary eigenvectors are orthogonal to all other boundary vectors y [except possibly when y∗ Ay is interior to a line segment in the boundary of W(A) through the given eigenvalue].

On atom-bond connectivity molecule structure descriptors

The atom-bond connectivity index ( ABC ) is a degree-based molecular structure descriptor with well-documented chemical applications. In 2010 a distance-based new variant of this index ( ABC GG ) has been proposed. Until now, the relation between ABC and ABC GG has not been analyzed. In this paper, we establish the basic characteristics of this relation. In particular, ABC and ABC GG are not correlated and both cases and may occur in the case of (structurally similar) molecules. However, in the case of benzenoid hydrocarbons, ABC always exceeds ABC GG .

Normalized Laplacian eigenvalues with chromatic number and independence number of graphs

ABSTRACT Let be the normalized Laplacian eigenvalues of a graph G with n vertices. Also, let χ and α be the chromatic number and the independence number of a graph G, respectively. In this paper, we discuss some properties of graphs with . In particular, we characterize all the graphs with when the maximum degree is n−1. Moreover, we obtain an upper bound on the multiplicity of normalized Laplacian eigenvalues in terms of n and α, and also characterize graphs for which the bound is attained.