The Laplacian spectra are the eigenvalues of Laplacian matrix
The graphs in this paper are simple and undirected. Let
The complement graph
Let
In the recent thirty years, the researchers obtained many good upper bounds for
At the end of this section, we introduce some lemmas which will be used later on.
Let
Moreover, if the row sums of
Let
The equality holds if and only if
This lemma gives a relation between the spectral radius of a graph and its line graph. Therefore, we can estimate the spectral radius of the adjacency matrix of graph by estimating that of its line graph.
Let
Here
Let
By these lemmas, we will give some improved upper bounds for the spectral radius and determine the corresponding extremal graphs.
This paper is organized as follows. In Section
The eigenvalues of adjacency matrix of the graph have wide applications in many fields. For instance, it can be used to present the energy level of specific electrons. Specially, the spectral radius of a graph is the maximum energy level of molecules. Hence, good upper bound for the spectral radius helps to estimate the energy level of molecules [
In the early time Cao [
The equality holds if and only if
Hu [
The equality holds if and only if
In 2005, Xu [
The equality holds if and only if
Using the average 2degree of the vertices, the researchers got more upper bounds.
Cao’s [
The equality holds if and only if
Similarly, Abrham and Zhang [
The equality holds if and only if
In recent years, Feng et al. [
The equality holds if and only if
The equality holds if and only if
The equality holds if and only if
The equality holds if and only if
All of these upper bounds mentioned in Section
In a graph, a circle with length 3 is called a triangle. If
Graph with triangles.
Let
Now, some new and sharp upper and lower bounds for the spectral radius will be given.
Let
Let
From Lemmas
It means that (
In a graph, let
Let
Since
Therefore, from Lemmas
Hence, it is easy to obtain that (
If equality in (
It means that
Similarly for the lower bound, if
It means that (
Let
According to the proof of Theorem
Thus
From Lemma
Solving this quadratic inequality, we obtain that upper bound (
If equality in (
In this section, we will present two graphs to illustrate that our some new bounds are better than other bounds in some sense. Let Figures
Graph of order 7.
Graph of order 8.
The estimated value of each upper bound is listed in Table
Estimated value of each upper bound.
Upper bounds  Figure 
Figure 

Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Bound ( 


Actual value 


In this part, we mainly discuss the upper bounds on the sum of Laplacian spectral radius of a connected graph
The following are some classic upper bounds of NordhausGaddum type. The coarse bound
In particular, if both
Liu et al. [
Shi [
To learn other bounds of the NordhausGaddum type, see references [
Here we give a new upper bound for the Laplacian spectral radius. For convenience, let
Let
Let
This is equivalent to the following inequality:
From Lemma
By simple calculation, we get the upper bound of the spectral radius of matrix
Since
If the spectral radius
Conversely, it is easy to verify that equality in (
In this part, based on Theorem
Let
According to the relation of a graph
Let
Then the upper bound of the NordhausGaddum type of Laplacian matrix is
since
Obviously,
Let
Here, the symbol
Simplifying this expression by direct calculation, we prove that the result (
If equality in (
In this section, we give some examples to illustrate that the new bound is better than other bounds for some graphs. Considering the graph of order 10 in Figure
Estimated value of each upper bound.
Upper bound  Figure 
Figure 
Figure 
Figure 

Bound ( 




Bound ( 




Bound ( 




Bound ( 




Bound ( 




Graph of order 10.
Clearly, from Table
From numerical examples of Sections
This work is supported by the Research Fund of Sichuan Provincial Education Department (Grant no. 11ZA159, 11ZZ020, 12ZB238, and 13ZB0108).