Affiliation:
1. Department of Applied Mathematics , Aligarh Muslim University , Aligarh , India
2. Department of Mathematics , University of Kashmir , Srinagar , India
Abstract
Abstract
Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as
L
E
=
∑
i
=
1
n
|
μ
i
-
2
m
n
|
LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|}
,
L
E
L
=
∑
i
=
1
n
-
1
μ
i
LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} }
and
K
f
=
n
∑
i
=
1
n
-
1
1
μ
i
Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}}
respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.