J Math Chem (2010) 47:21–28
DOI 10.1007/s10910-009-9529-1
ORIGINAL PAPER
Maximum eigenvalue of the reciprocal distance matrix
Kinkar Ch. Das
Received: 10 December 2008 / Accepted: 21 January 2009 / Published online: 12 February 2009
© Springer Science+Business Media, LLC 2009
Abstract 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.
Keywords Molecular graph · Reciprocal distance matrix · Maximum eigenvalue ·
Diameter · Lower bound · Upper bound
1 Introduction
Since the distance matrix and related matrices, based on graph-theoretical distances
[1], are rich sources of many graph invariants (topological indices) that have found
use in structure-property-activity modeling [2–4], it is of interest to study spectra and
polynomials of these matrices [5–7].
Let G = (V , E ) be a simple connected graph with vertex set V (G) ={v
1
,v
2
,...,
v
n
} and edge set E (G), where |V (G)|= n and | E (G)|= m. Let G be the complement
of G. For v
i
∈ V (G), Ŵ(v
i
) denotes the set of its (first) neighbors in G and the degree
of v
i
is d
i
=|Ŵ(v
i
)|. The minimum vertex degree is denoted by δ and the maximum
by . The average of the degrees of the vertices adjacent to v
i
is denoted by μ
i
. The
diameter of a graph is the maximum distance between any two vertices of G. Let d be
the diameter of G. The distance matrix D of G is an n × n matrix (d
i , j
) such that d
i , j
is just the distance (i.e., the number of edges of a shortest path) between the vertices
v
i
and v
j
in G [1]. The reciprocal distance matrix RD of G, also called the Harary
matrix [1], is an n × n matrix ( RD
i , j
) such that [8, 9].
K. Ch. Das (B )
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
e-mail: kinkar@lycos.com
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