1 Introduction

Randić energy of specific graphs


Saeid Alikhani111Corresponding author. E-mail: alikhani@yazd.ac.irand Nima Ghanbari

Department of Mathematics, Yazd University

89195-741, Yazd, Iran

  Abstract

Let be a simple graph with vertex set . The Randić matrix of , denoted by , is defined as the matrix whose -entry is if and are adjacent and for another cases. Let the eigenvalues of the Randić matrix be which are the roots of the Randić characteristic polynomial . The Randić energy of is the sum of absolute values of the eigenvalues of . In this paper we compute the Randić characteristic polynomial and the Randić energy for specific graphs .

Mathematics Subject Classification: 15A18, 05C50.
Keywords: Randić matrix; Randić energy; Randicć characteristic polynomial; eigenvalues.

 

1 Introduction

In this paper we are concerned with simple finite graphs, without directed, multiple, or weighted edges, and without self-loops. Let be such a graph, with vertex set . If two vertices and of are adjacent, then we use the notation . For , the degree of the vertex , denoted by , is the number of the vertices adjacent to .

Let be adjacency matrix of and its eigenvalues. These are said to be 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 [6, 7, 9, 10].

In 1975 Milan Randić invented a molecular structure descriptor defined as [11]

The Randić-index-concept suggests that it is purposeful to associate to the graph a symmetric square matrix . The Randić matrix is defined as [2, 3, 8]

Denote the eigenvalues of the Randić matrix by and label them in non-increasing order. Similar to characteristic polynomial of a matrix, we consider the Randić characteristic polynomial of (or a graph ), as which is equal to . The Randić energy [2, 3, 8] of is defined as

For several lower and upper bounds on Randić energy, see [2, 3, 8].

In Section 2, we obtain the Randić characteristic polynomial and energy of specific graphs. As a result, we show that for every natural number , there exists a graph such that . In Section 3, we find Randić energy of specific graphs with one edge deleted.

As usual we denote by a matrix all of whose entries are equal to .

2 Randić characteristic polynomial and Randić energy of specific graphs

In this section we study the Randić characteristic polynomial and the Randić energy for certain graphs. The following theorem gives a relationship between the Randić energy and energy of path .

Lemma 1

[8] Let be the path on vertices. Then

The following theorem gives the Randić energy of even cycles.

Lemma 2

[12] Let be the cycle on vertices for . Then

Here we shall compute the Randić characteristic polynomial of paths and cycles.

Theorem 1

. For , the Randić characteristic polynomial of the path graph satisfy

where for every , with and .

Proof. For every , consider

and let . It is easy to see that .

Suppose that . We have

So

And so

Therefore

and so

Hence

Theorem 2

. For , the Randić characteristic polynomial of the cycle graph is

where for every , with and .

Proof. Similar to the proof of Theorem 1, for every , we consider

and let . We have .

Suppose that . We have

So

And so,

Therefore,

Hence

In [12] has shown that the Randić energy of , the star on vertices and the complete bipartite graph is . Here using the Randić characteristic polynomial, we prove these results. We need the following lemma:

Lemma 3

. [4] If is a nonsingular square matrix, then

Theorem 3

. For ,

  • The Randić characteristic polynomial of the star graph is

  • The Randić energy of is

Proof.

  1. It is easy to see that the Randić matrix of is . We have

    Using Lemma 3,

    We know that . Therefore

    Since the eigenvalues of are (once) and 0 ( times), the eigenvalues of are (once) and 0 ( times). Hence

  2. It follows from Part (i).      

Theorem 4

. For ,

  • the Randić characteristic polynomial of complete graph is

  • the Randić energy of is

Proof.

  1. It is easy to see that the Randić matrix of is . Therefore

    Since the eigenvalues of are (once) and 0 ( times), the eigenvalues of are (once) and 0 ( times). Hence

  2. It follows from Part (i).      

Theorem 5

. For natural number ,

  • The Randić characteristic polynomial of complete bipartite graph is

  • The Randić energy of is

Proof.

  1. It ie easy to see that the Randić matrix of is . Using Lemma 3 we have

    So

    We know that . Therefore

    The eigenvalues of are (once) and 0 ( times). So the eigenvalues of are (once) and 0 ( times). Hence

  2. It follows from Part (i).      

Let be any positive integer and be friendship graph with vertices and edges. In other words, the friendship graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. The Friendship Theorem of Erdős, Rényi and Sós [5], states that graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. The Figure 1 shows some examples of friendship graphs. Here we shall investigate the Randić energy of friendship graphs.

Figure 1: Friendship graphs and , respectively.
Theorem 6

. For ,

  • The Randić characteristic polynomial of friendship graph is

  • The Randić energy of is

Proof.

  1. The Randić matrix of is

    Now for computing , we consider its first row. The cofactor of the first array in this row is

    and the cofactor of another arrays in the first row are similar to

    Now, by straightforward computation we have the result.

  2. It follows from Part (i).       

Remark. In [1] has shown that the energy of a graph cannot be an odd integer. Since for , the Randić energy can be odd or even integer. More precisely we have:

Corollary 1

. For every natural number , there exists a graph such that .

Proof. If then consider and for we consider friendship graphs .      

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. The Figure 2 shows some examples of Dutch Windmill graphs. Here we shall investigate the Randić energy of Dutch Windmill graphs.

Figure 2: Dutch Windmill Graph and , respectively.
Theorem 7

. For ,

  • The Randić characteristic polynomial of friendship graph is

  • The Randić energy of is

Proof.

  1. The Randić matrix of is

    Let , and .

    Then

    Now, by the straightforward computation we have the result.

  2. It follows from Part (i).      

3 Randić energy of specific graphs with one edge deleted

In this section we obtain the randić energy for certain graphs with one edge deleted. We need the following lemma:

Lemma 4

. Let . Then

Here we state the following easy results:

Lemma 5

.

  1. If , then , where .

  2. If , (), then

  3. Let be the star on vertices and . Then for any ,

Theorem 8

. For ,

  • Let be an edge of complete graph . The Randić characteristic polynomial of is

  • The Randić energy of is

Proof.

  1. The Randić matrix of is

    Therefore

    Similar to the proof of Theorem 5(i) we have the result.

  2. It follows from Part (i).      

Theorem 9

. For ,

  • The Randić characteristic polynomial of complete bipartite graph is

  • The Randić energy of is

Proof.

  1. The Randić matrix of is

    where .So

    Similar to the proof of Theorem 5(i) we have the result.

  2. It follows from Part (i).      

References

  • [1] R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc., 1 (2004), 129–132.
  • [2] Ş. B. Bozkurt, A. D. Güngör, I. Gutman, A. S. Çevik, Randić matrix and Randić energy, MATCH Commum. Math. Comput. Chem. 64 (2010) 239–250.
  • [3] Ş. B. Bozkurt, A. D. Güngör, I. Gutman, Randić spectral radius and Randić energy, MATCH Commum. Math. Comput. Chem. 64 (2010) 321–334.
  • [4] D. Cvetković, M. Doob, H. sachs, Spectra of graphs - Theory and Aplication, Academic Press, New York, 1980.
  • [5] P. Erdös, A. Rényi, V.T. Sós, On a problem of graph theory, Studia Sci. Math. Hungar., 1, 215–235 (1966).
  • [6] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A.Kohnert, R. Laue, A. Wassermannn (Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, pp. 196–211.
  • [7] I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total -electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441–456.
  • [8] I. Gutman, B. Furtula, Ş. B. Bozkurt, On Randić energy, Linear Algebra Appl., 442 (2014) 50–57.
  • [9] I. Gutman, X. Li, J. Zhang, Graph energy, in: M. Dehmer, F. Emmert-Streib (Eds.), Analysis of Complex Networks. From Biology to Linguistics, Wiley-VCH, Weinheim, 2009, pp. 145–174.
  • [10] S. Majstorović, A. Klobu·car, I. Gutman, Selected topics from the theory of graph energy: hypoenergetic graphs, in: D. Cvetković, I. Gutman (Eds.), Applications of Graph Spectra, Math. Inst., Belgrade, 2009, pp. 65–105.
  • [11] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609–6615.
  • [12] O. Rojo, L. Medina, Construction of bipartite graphs having the same Randić energy, MATCH Commun. Math. Comput. Chem. 68 (2012) 805–814.
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