More on energy and Randić energy of specific graphs
Saeid Alikhani^{1}^{1}1Corresponding author. Email: alikhani@yazd.ac.irand Nima Ghanbari^{2}^{2}2Email: n.ghanbari@stu.yazd.ac.ir
Department of Mathematics, Yazd University
89195741, Yazd, Iran
Abstract
Let be a simple graph of order . The energy of the graph is the sum of the absolute values of the eigenvalues of . The Randić matrix of , denoted by , is defined as the matrix whose entry is if and are adjacent and for another cases. The Randić energy of is the sum of absolute values of the eigenvalues of . In this paper we compute the energy and Randić energy for certain graphs. Also we propose a conjecture on Randić energy.
Mathematics Subject Classification: 15A18.
Keywords: 3regular graphs; energy; Randić energy; characteristic polynomial; Petersen graph.
1 Introduction
In this paper we are concerned with simple finite graphs, without directed, multiple, or weighted edges, and without selfloops. Let be adjacency matrix of and its eigenvalues. These are said to be the eigenvalues of the graph and to form its spectrum [4]. The energy of the graph is defined as the sum of the absolute values of its eigenvalues
Details and more information on graph energy can be found in [5, 6, 8, 12].
The Randić matrix is defined as [2, 3, 9]
Denote the eigenvalues of the Randić matrix by and label them in nonincreasing order. The Randić energy [2, 3, 9] of is defined as
Two graphs and are said to be Randić energy equivalent, or simply equivalent, written , if . It is evident that the relation of being equivalence is an equivalence relation on the family of graphs, and thus is partitioned into equivalence classes, called the equivalence classes. Given , let
We call the equivalence class determined by . A graph is said to be Randić energy unique, or simply unique, if .
Similarly, we can define equivalence for energy and unique for a graph.
A graph is called regular if all vertices have the same degree . One of the famous graphs is the Petersen graph which is a symmetric nonplanar 3regular graph. In the study of energy and Randić energy, it is interesting to investigate the characteristic polynomial and energy of this graph. We denote the Petersen graph by .
In this paper, we study the energy and Randić energy of specific graphs. In the next section, we study energy and Randić energy of regular and regular graphs. We study cubic graphs of order 10 and list all characteristic polynomial, energy and Randić energy of them. As a result, we show that Petersen graph is not unique (unique) but can be determined by its Randić energy (energy) and its eigenvalues. In the last section we consider some another families of graphs and study their Randić characteristic polynomials.
2 Energy of 2regular and 3regular graphs
The energy and Randić energy of regular graphs have not been widely studied. In this section we consider regular and regular graphs. The following theorem gives a relationship between the Randić energy and energy of regular graphs.
Lemma 1
[10] If the graph is regular then .
Also we have the following easy lemma:
Lemma 2
. Let . Then

.

.
Randić characteristic polynomial of the cycle graph can be determined by the following theorem:
Lemma 3
By Lemma 3, we can find all the eigenvalues of Randić matrix of cycle graphs. So we can compute the Randić energy of cycles. Also every cycle is regular. By Lemma 1, we have . Hence we can compute energy of cycle graphs too. Every regular graph is a disjoint union of cycles. Therefore by Lemma 2, we can find energy and Randić energy of 2regular graphs.
Let to consider the characteristic polynomial of regular graphs of order . Also we shall compute energy and Randić energy of this class of graphs. There are exactly cubic graphs of order given in Figure 1 (see [11]).
We show that Petersen graph is not unique (unique) but can be determined by its Randić energy (energy) and its eigenvalues. There are just two nonconnected cubic graphs of order . The following theorem gives us characteristic polynomial of regular graphs of order . We denote the characteristic polynomial of the graph by .
Using Maple we computed the characteristic polynomials of regular graphs of order in Table 1.
Table 1. Characteristic polynomial , for .
By computing the roots of characteristic polynomial of cubic graphs of order , we can have the energy of these graphs. We compute them to four decimal places. So we have table 2:
15.1231  5.0410  15.1231  5.0410  14.7943  4.9314  
14.8596  4.9532  15.3164  5.1054  14.0000  4.6666  
14.8212  4.9404  14.4721  4.8240  16.0000  5.3333  
13.5143  4.5047  14.7020  4.9006  13.5569  4.5189  
14.2925  4.7641  16.0000  5.3333  15.5791  5.1930  
14.9443  4.9814  14.3780  4.7926  14.0000  4.6666  
15.0777  5.0259  15.0895  5.0298  12.0000  4.0000 
Table 2. Energy and Randić energy of cubic graphs of order .
Theorem 1
. Six cubic graphs of order are not unique (unique). If two cubic graphs of order have equal energy (Randić energy), then their eigenvalues are different in exactly values.
Proof. Using Table 2, we see that , and . Now, it suffices to find the eigenvalues of , , , , and . By Table 1 we have:
Also
And
So we have the result.
Now we consider Petersen graph . We have shown this graph in Figure 2.
Theorem 2
. Let be the family of regular graphs of order . For the Petersen graph , we have the following properties:

is not unique (unique) in .

has the maximum energy (Randić energy) in .

can be identify by its energy (Randić energy) and its eigenvalues in .
Proof.

The adjacency matrix of is
So Therefore we have:
and so we have . By Table 2, we have . Hence is not unique (and unique) in .

It follows from Part (i) and Table 2.

It follows from Part (i) and Theorem 1. So is the Petersen graph.
The following result gives a relationship between energy and permanent of adjacency matrix of two connected graphs in the family of cubic graphs of order whose have the same equivalence class.
Theorem 3
. If two connected cubic graphs of order have the same energy, then their adjacency matrices have the same permanent.
Proof. By Table 2, it suffices to find , , and . For graph , we have
By Ryser’s method, we have . Similarly we have:
So we have the result.
Remark 1. The converse of Theorem 4 is not true. Because , but as we see in Table 2, .
Corollary 1
. If two connected cubic graphs of order have the same Randić energy, then their adjacency matrices have the same permanent.
3 Randić characteristic polynomial of a kind of DutchWindmill graphs
We recall that a complex number is called an algebraic number (resp. an algebraic integer) if it is a root of some monic polynomial with rational (resp. integer) coefficients (see [13]). Since the Randić characteristic polynomial is a monic polynomial in with integer coefficients, its roots are, by definition, algebraic integers. This naturally raises the question: Which algebraic integers can occur as zeros of Randić characteristic polynomials? And which real numbers can occur as Randić energy of graphs? We are interested to numbers which are occur as Randić energy. Clearly those lying in are forbidden set, because we know that if graph possesses at least one edge, then . We think that the Randić energy of graphs are dense in . In this section we would like to study some further results of this kind.
Let be any positive integer and be Dutch Windmill graph with vertices and edges. In other words, the graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. We recall that is friendship graphs. Figure 3 shows some examples of this kind of Dutch Windmill graphs. In this section we shall investigate the Randić characteristic polynomial of Dutch Windmill graphs.
By Lemma 3, we know that
where for every , with and . We show that the Randić characteristic polynomial of Dutch Windmill graphs can compute by the cycle which constructed it.
Theorem 4
. For , the Randić characteristic polynomial of the Dutch Windmill graph is
where for every , with and .
Proof. For every , consider
and let . It is easy to see that .
Suppose that . We have
where . So
Therefore
Hence
In [1] we have presented two families of graphs such that their Randić energy are and . Here we recall the following results:
Theorem 5
. [1]

The Randić energy of friendship graph is

The Randić energy of DutchWindmill graph is

For every , the Randić energy of is
We can use Theorem 4 to obtain . Here using the definition of Randić characteristic polynomial, we prove the following result:
Theorem 6
. The Randić energy of is
Proof. The Randić matrix of is
Let and . Then
So
Hence
Part (iii) of Theorem 5 implies that the Randić energy of graphs are dense in . Motivated by this notation, Theorems 5 and 6, we think that the Randić energy of graphs are dense in . We close this paper by the following conjecture:
Conjecture 1
. Randić energy of graphs are dense in .
References
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