On Spectral Radius of Graphs

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518

1

On Spectral Radius of Graphs 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐺, 𝑅𝑅𝑅𝑅𝑅𝑅 𝐾𝐾𝐾𝐾𝐾 𝑀. 𝑅, 𝑁𝑁𝑁𝑁𝑁 𝐾𝐾𝐾𝐾𝐾 V.

Abstract—Let G(V, E) be simple graph with n vertices and m edges and A be vetex subset of V(G ). For any v∈ 𝐴 the degree of the vertex 𝑣𝑖 with respect to the subset A is defined as the number of vertices A that are adjacent to 𝑣𝑖 . We call it as 𝔇 −degree and is denoted by 𝔇𝑖 . Denote 𝜆1 (𝐺) as the largest eigenvalue of the graph G and 𝑠𝑖 .as the sum of 𝔇 degree of vertices that are adjacent to 𝑣𝑖 . In this paper we give lower bounds of 𝜆1 (𝐺) in terms of 𝔇 degree. Index Terms—𝔇-degree, spectral radius, dominating set, covering set, adjacency matrix.

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1 INTRODUCTION

L

et G be simple graph with n vertices and m edges. For any 𝑣𝑖 ∈ 𝑉, the degree of 𝑣𝑖 , denoted by 𝑑𝑖 , is the number of edges that are adjacent to 𝑣𝑖 . The new definition of the degree of a vertex and results using this can be seen in papers[8, 9]. A subset D of V is called dominating set if every vertex of V-D is adjacent to to some vertex in D. Any dominating set with minimum cardinality is called minimum dominating set and this cardinality is called domination number. A subset C of V is called covering set if every edge of V is incident to at least one vertex of C. Any covering set with minimum cardinality is called minimum covering set and the number of element of this set is called covering number.

Here A may or may not be the subset of V. However, if A=∅ then 𝔇𝑖 =0 and if A=V then 𝔇𝑖 = d(𝑣𝑖 ) ∀ 𝑣𝑖 ∈ 𝑉.

2

BOUNDS FOR SPECTRAL RADIUS

Theorem 2.1. Let G be a connected graph with n vertices as 𝑣1 , 𝑣2 , … , 𝑣𝑛 . If 𝔇1 , 𝔇2 , . . . , 𝔇𝑛 represents 𝔇-degree sequence of these vertices with respect to a vertex subset A of G then

IJSER

For any graph G, let A(G)=(𝑎𝑖𝑗 ) be the adjacency matrix. The eigenvalues 𝜆1 , 𝜆2 , … , 𝜆𝑛 of A assumed in non-increasing order. The largest eigenvalue 𝜆1 (𝐺) is called spectral radius of the graph. Till now, there are many bounds for 𝜆1 (𝐺 ) (see for instance [3,6]). Further results and properties see the reviews [1,2,4,5]. In this paper, we give bounds for 𝜆1 of G in tems of 𝔇-degree, which is generalization of earlier bounds found in the paper [3,6]. Using minimum dominating set and minimum covering set, the adjacency matrices 𝐴𝐷 (𝐺) , 𝐴𝐶 (𝐺) have been defined in the papers [7,10]. The 𝔇-degree is also used to get bounds for the largest eigenvalue 𝜆𝐷1 (𝐺), 𝜆1𝐶 (𝐺) of these matrices. We start with defining degree of vertex with respect to a subset of vertex set or 𝔇-degree. Definition1.1: Let G(V, E) be a simple graph and A be any vertex sub-set. The degree of a vertex 𝑣𝑖 of a graph G with respect to A is the number of vertices of A that are adjacent to 𝑣𝑖 . This degree is denoted by 𝑑𝐴 (𝑣𝑖 ) 𝑜𝑜 𝔇𝑖 .

𝜆1 (𝐺) ≥ �

Where, s i is sum of 𝔇 degree vertices that are adjacent to v i . Equality case occurs when A=V

Proof. Let X = (𝑥1 , 𝑥2 , … , 𝑥𝑛 )𝑇 be a unit positive eigenvector of A(G) corresponding to 𝜆1 (G). Let C = �∑𝑛

• 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐺, 𝑅𝑅𝑅𝑅𝑅𝑅 𝐾𝐾𝐾𝐾𝐾 𝑀. 𝑅, Post Graduate Department of Mathematics, Maharani’s Science College for Women, J. L. B Road, Mysore-570005, India. E-mail : [email protected], [email protected] • 𝑁𝑁𝑁𝑁𝑁 𝐾𝐾𝐾𝐾𝐾 V. Assistant Professor of Physics, Government College for Women, Hole Narasipura, Hassan, Indias E-mail : [email protected]

1

2 𝑖=1 𝔇𝑖

AC = �∑𝑛

1

2 𝑖=1 𝔇𝑖

= �∑𝑛

1

2 𝑖=1 𝔇𝑖

(𝔇1 , 𝔇2 , … , 𝔇𝑛 )𝑇

(∑𝑛𝑗=1 𝑎1𝑗 𝔇𝑗 , … , ∑𝑛𝑗=1 𝑎𝑛𝑗 𝔇𝑗 )𝑇 (𝑠1 , 𝑠2 , … , 𝑠𝑛 )𝑇

Similarly 𝐶 𝑇 𝐴 = �∑𝑛 1

————————————————

𝑠12 + 𝑠22 + … + 𝑠𝑛2 𝔇12 + 𝔇22 + … + 𝔇2𝑛

𝐶 𝑇 𝐴2 𝐶= ∑𝑛

2 1 𝔇𝑖

1

2 𝑖=1 𝔇𝑖

(𝑠1 , 𝑠2 , … , 𝑠𝑛 )

(𝑠12 + 𝑠22 + ⋯ + 𝑠𝑛2 )

∴ 𝜆1 (𝐺)= �𝜆1 (𝐴2 ) = √𝑋 𝑇 𝐴2 𝑋 ≥ √𝐶 𝑇 𝐴2 𝐶 Hence 𝜆1 (𝐺) ≥ �

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2 𝑠12 +𝑠22 +…+𝑠𝑛

2 2 𝔇2 1 +𝔇2+…+𝔇𝑛

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518

Corollary 2.1. When A= V, 𝜆1 (𝐺) ≥ �

2 𝑡12 +𝑡22 +…+𝑡𝑛

2 2 𝔇2 1 +𝔇2 +…+𝔇𝑛

where 𝑡𝑖 represents the sum of degree of vertices that are adjacent to 𝑣𝑖 . Theorem 2.2. Let G be a connected graph with 𝔇-degree sequence of vertices as 𝔇1 , 𝔇2 , 𝔇3 , … , 𝔇𝑛 corresponding to the subset A of G then

(𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛 )2 𝜆1 (𝐺) ≥ � 𝑛(𝔇12 + 𝔇22 + … + 𝔇2𝑛 )

Proof. By theorem2.1, 𝜆1 (𝐺 ) ≥ �

2 𝑠12+𝑠22 +…+𝑠𝑛

By Cauchy’s Schwarz inequality,∑𝑛𝑖=1 𝑠𝑖2 ≥ ∴𝜆1 (𝐺) ≥ �

2 (∑𝑛 𝑖 𝑠𝑖)

2 2 𝑛(𝔇2 1 +𝔇2 +…+𝔇𝑛 )

Also 𝑠1 + 𝑠2 + ⋯ + 𝑠𝑛 = 𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛

∑𝑘 𝑖=1 𝑑𝑖 𝑛

Let C = �∑𝑛

(𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛 )2 𝜆1 (𝐺) ≥ � 𝑛(𝔇12 + 𝔇22 + … + 𝔇2𝑛 )

Since 𝑑𝑖 ≥ 𝔇𝑖 , 𝜆1 (𝐺 ) ≥ �

2 ∑𝑛 𝑖=1 𝔇𝑖

(D+A)C=�∑𝑛 = �∑𝑛

𝑛2

𝜆1 (𝐺 ) ≥

1

2 𝑖=1 𝔇𝑖

Similarly , 2 (∑𝑛 𝑖 𝔇𝑖 )

(𝔇1+𝔇2 +⋯+𝔇𝑛 )2

where k = |D|

(𝔇1 , 𝔇2 , … , 𝔇𝑛 )𝑇

(∑𝑛𝑖=1(𝑑1𝑗 + 𝑎1𝑗 )𝔇𝑗 , … , ∑𝑛𝑗=1(𝑑𝑛𝑗 + 𝑎𝑛𝑗 )𝔇𝑗 )𝑇

(𝑠1 + ∑𝑛𝑗=1 𝑑1𝑗 𝔇𝑗 , … , 𝑠𝑛 + ∑𝑛𝑗=1 𝑑𝑛𝑗 𝔇𝑗 )𝑇 𝑛

𝑛

𝑗=1

𝑗=1

1 𝐶 𝑇 (𝐷 + 𝐴) = � 𝑛 �𝑠1 + � 𝑑𝑖𝑗 𝔇𝑗 , … , 𝑠𝑛 + � 𝑑𝑛𝑗 𝔇𝑗 � ∑𝑖=1 𝔇2𝑖

𝑛

∑𝑛𝑖=1 𝔇𝑖

Therefore 𝐶 𝑇 (𝐷 + 𝐴)2 𝐶 =

𝑛

But ∑𝑛𝑖=1 𝔇𝑖 = ∑𝑘𝑖=1 𝑑𝑖 where k = |A| 𝜆1 (𝐺 ) ≥ (𝑛−1) 𝑛

.

𝑛

2

𝑛

2

1 𝑛 2 ��𝑠1 + � 𝑑𝑖𝑗 𝔇𝑗 � + ⋯ + �𝑠𝑛 + � 𝑑𝑛𝑗 𝔇𝑗 � � ∑1 𝔇𝑖

∑𝑘 𝑖=1 𝑑𝑖

Corollary 2.3. If A = D , a minimum dominating set, Then ∑𝑘𝑖=1 𝑑𝑖 ≥ (𝑛 − 1) and so 𝜆1 (𝐺 ) ≥

1

2 𝑖=1 𝔇𝑖

(𝔇21 +𝔇22 +⋯+𝔇2𝑛 ) 𝑛

1

2 𝑖=1 𝔇𝑖

(𝔇21 +𝔇22 +⋯+𝔇2𝑛 )2 𝑛(𝔇21+𝔇22+…+𝔇2𝑛)

By Cauchy’s Scwarz inequality, ∑𝑛𝑖=1 𝔇2𝑖 ≥ Thus 𝜆1 (𝐺 ) ≥ �

2 𝑘 2 ∑𝑛 𝑖=1 𝑠𝑖 + ∑𝑖=1 𝔇𝑖

IJSER

subset A with |A|= k then 𝜆1 (𝐺 ) ≥

≥�

1 𝑖𝑖 𝑖 = 𝑗 𝑎𝑎𝑎 𝑣𝑖 ∈ 𝐷 0 𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒

Where, s i is sum of 𝔇 degree vertices that are adjacent to v i . Equality case occurs when D =V. Proof. Let X = (𝑥1 , 𝑥2 , … , 𝑥𝑛 )𝑇 be a unit positive eigenvector of 𝐴𝐷 (𝐺) corresponding to 𝜆𝐷 1 (𝐺)).

Corollary 2.2. For a connected graph G and a vertex

Which implies 𝜆1 (𝐺 )

Definition 2.1. The minimum dominating adjacency matrix 𝐴𝐷 (𝐺)is defined by 𝐴𝐷 (𝐺) = 𝐷(𝐺)+ A(G) , where A(G) is the adjacency matrix and D(G ) =(𝑑𝑖𝑗 ) is 𝑛 × 𝑛 matrix with

𝜆𝐷 1 (𝐺) ≥ �

(𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛 )2 𝜆1 (𝐺) ≥ � 𝑛(𝔇12 + 𝔇22 + … + 𝔇2𝑛 )

Proof: By theorem 2.2

We recall the definition of minimum dominating matrix 𝐴𝐷 (𝐺), as defined in [7].

Theorem 2. 3. Let G be a connected graph with n vertices and D be a minimum dominating set. If 𝔇1 , 𝔇2 , . . . , 𝔇𝑛 represents 𝔇-degree sequence of these vertices corresponding to the vertex subset D of G and 𝐴𝐷 (𝐺) is minimum dominating matrix then

𝑛

(𝑠1 +𝑠2 +⋯+𝑠𝑛 )2

Corollary 2.4. If A=V then ∑𝑛𝑖=1 𝔇𝑖 = ∑𝑛𝑖=1 𝑑𝑖 = 2𝑚 2𝑚 therefore, 𝜆1 (𝐺 ) ≥ 𝑛 .

𝑑𝑖𝑗 =�

2 2 𝔇2 1 +𝔇2 +…+𝔇𝑛

2

𝑛

𝑗=1

≥

1

2 ∑𝑛 1 𝔇𝑖

2

[𝑠12 + 𝑠22 … + 𝑠𝑛2 +(∑𝑛𝑗=1 𝑑1𝑗 𝔇𝑗 )2 + ⋯ + �∑𝑛𝑗=1 𝑑𝑛𝑗 𝔇𝑗 � ]

Hence 𝐶 𝑇 (𝐷 + 𝐴)2 𝐶 ≥

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𝑗=1

1

2 ∑𝑛 1 𝔇𝑖

[𝑠12 + 𝑠22 … + 𝑠𝑛2 + ∑𝑘𝑖=1 𝔇2𝑖 ]

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518

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where k = |D|

∴ 𝜆1𝐷 (𝐺)= �𝜆1 (𝐷 + 𝐴)2 = �𝑋 𝑇 (𝐷 + 𝐴)2 𝑋 ≥ �𝐶 𝑇 (𝐷 + 𝐴)2 𝐶

∴

𝜆1𝐷 (𝐺)

𝑛

𝑘 ∑𝑛 𝑠 2 + ∑𝑖=1 𝔇2𝑖 ≥ � 𝑖=1 𝑖 𝑛 2 ∑𝑖=1 𝔇𝑖

2 𝑘 (𝑑1 𝔇1 +𝑑2 𝔇2 +⋯+𝑑𝑛 𝔇𝑛 + ∑𝑖=1 𝔇𝑖 )

𝜆𝐷 1 (𝐺) ≥ �

�� 𝔇2𝑖 𝑖=1

2 ∑𝑛 𝑖=1 𝔇𝑖

�𝑛 ∑𝑛𝑖=1 𝔇2𝑖

𝑘 (∑𝑛 𝑖=1 𝔇𝑖 + ∑𝑖=1 𝔇𝑖 )

𝑛

∑𝑘𝑖=1(𝑑𝑖 + 𝔇𝑖 ) 𝑛

𝜆1𝐷 (𝐺) ≥

𝜆1𝐷 (𝐺) ≥

2𝑚 + 𝑘 𝑛

Proof. Since 𝑑𝑖 ≥ 𝔇𝑖 , we get new bounds for 𝜆1𝐷 (𝐺) (∑𝑛𝑖=1 𝔇2𝑖 + ∑𝑘𝑖=1 𝔇2𝑖 ) �𝑛 ∑𝑛𝑖=1 𝔇2𝑖 𝑎+𝑏

But for any real numbers a and b

(∑𝑛𝑖=1 𝔇2𝑖 + ∑𝑘𝑖=1 𝔇2𝑖 )

≥

√𝑎

≥ �(𝑎 + 𝑏) so

𝑛

�(� 𝔇2𝑖 𝑖=1

+

𝑘

𝑘

𝑘

𝑖=1

𝑖=1

� 𝔇𝑖 = � 𝑑𝑖 ≥ 𝑘

Therefore 𝜆𝐷 1 (𝐺 ) ≥

2𝑚+𝑘 𝑛

The above results is true for minimum covering matrix [10]. Hence we have the following theorem.

Theorem 2.5. Let G be a connected graph and D represents minimum dominating set with k = |D| then

𝑛 𝔇2𝑖 �∑𝑖=1

𝜆𝐷 1 (𝐺 ) ≥

IJSER

𝑘 (∑𝑛𝑖=1 𝑠𝑖2 + ∑𝑖=1 𝔇2𝑖 )2 𝑛 2 𝑛 ∑𝑖=1 𝔇𝑖

2 (𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛 + ∑𝑘 𝑖=1 𝔇𝑖 )

𝜆𝐷1 (𝐺) ≥

𝑖=1

2

When D = V, ∑𝑛𝑖=1 𝔇𝑖 = ∑𝑛𝑖=1 𝑑𝑖 =2m and

2 𝑘 2 ∑𝑛 𝑖=1 𝑠𝑖 + ∑𝑖=1 𝔇𝑖

Also 𝑠1 + 𝑠2 + ⋯ + 𝑠𝑛 = 𝑑1 𝔇1 + 𝑑2 𝔇2 + ⋯ + 𝑑𝑛 𝔇𝑛

𝜆𝐷 1 (𝐺) ≥

(∑𝑛𝑖=1 𝔇𝑖 + ∑𝑘𝑖=1 𝔇𝑖 ) 𝑛

+ � 𝔇2𝑖 � ≥

Since ∑𝑛𝑖=1 𝔇𝑖 = ∑𝑘𝑖=1 𝑑𝑖 where k = |D|

By Cauchy Schwarz inequality, the above relation simplifies to 𝜆1𝐷 (𝐺) ≥ �

𝑘

The above relations simplifies to

where k = |D|

�𝑛 ∑𝑛𝑖=1 𝔇2𝑖

Proof. By theorem 2.3

√𝑛

Using Cauchy Schawarz inequality

Theorem 2. 4. Let G be a connected graph with n vertices and D be minimum dominating set. If 𝔇1 , 𝔇2 , . . . , 𝔇𝑛 represents 𝔇-degree sequence of these vertices corresponding to the vertex subset D of G and 𝐴𝐷 (𝐺)is minimum dominating matrix then

𝜆𝐷 1 (𝐺) ≥

�(∑𝑛𝑖=1 𝔇2𝑖 + ∑𝑘𝑖=1 𝔇2𝑖 )

𝜆𝐷 1 (𝐺) ≥

Theorem2.6. Let G be a connected graph and C be a minimum covering set. The minimum adjacency covering matrix, 𝐴𝐶 (𝐺) defined by

𝐴𝐶 (𝐺) = 𝐷(𝐺)+ A(G) , where A(G) is the adjacency matrix and D(G ) =(𝑑𝑖𝑗 ) is 𝑛 × 𝑛 matrix with 1 𝑖𝑖 𝑖 = 𝑗 𝑎𝑎𝑎 𝑣𝑖 ∈ 𝐶 𝑑𝑖𝑗 =� 0 𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒

Then 𝜆𝐶 1 (𝐺) ≥ � 𝜆1𝐶 (𝐺) ≥

2 𝑐 2 ∑𝑛 𝑖=1 𝑠𝑖 + ∑𝑖=1 𝔇𝑖 2 ∑𝑛 𝑖=1 𝔇𝑖

(𝑑1 𝔇1 +𝑑2 𝔇2 +⋯+𝑑𝑛𝔇𝑛+ ∑𝑐𝑖=1 𝔇2 𝑖) 2 �𝑛 ∑𝑛 𝑖=1 𝔇𝑖

𝜆1𝐷 (𝐺) ≥

where, c is covering number

� 𝔇2𝑖 ) 𝑖=1

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where c = |C|

2𝑚 + 𝑐 𝑛

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518

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CONCLUSION

In this paper we obtained lower bounds for largest eigen value(Spectral radius) in terms of 𝔇-degree. It was proved that these bounds are generalization of earlier bounds found in the papers[5,6].

REFERENCES D. Cvetkovic, I. Gutman, eds, “Applications of Graph Spectra,” Mathematical Institution, Belgrade, 2009 [2] D. Cvetkovic, I. Gutman, eds, Selected Topics on Applications of Graph Spectra, Mathematical Institution, Belgrade, 2011 [3] H. S. Ramane, H. B. Walikar, “Bounds for the eigenvalues of a graph, Graphs, Combinatorics, algorithms and applications,” Narosa publishing House, New Delhi, 2005 [4] D. Cvetkovic, M. Doob, H. Sachs, “Spectra of graphs – Theory and Application,"Academic Press, Newyork, 1980 [5] M. Hofmeister, "Spectral radius and degree sequence, " Math. Nachr,vol. 139, pp. 37-44, 1988 [6] A. Yu, M. Lu, F. Tian, “On Spectral radius of graphs,” Linear algebra and its application, vol. 387, pp. 41-49, 2004 [7] M. R. Rajesh Kanna, B. N. Dharamendra, G. Sridhara, “ Minimum dominating energy of a graphs,” International Journal of Pure and Applied Mathematics , vol. 85(4), pp. 707 - 718, 2013 [8] M. R. Rajesh Kanna, B. N. Dharamendra, G. Sridhara, “ Some results on the degree of a vertex of graph with respect to any vertexset,” International Journal of Contemporary Mathematical Sciences, vol. 8, pp. 1-4, 2013 [9] S. S. Kamath and R. S. Bhat, “Some new degree concepts in graphs’” Proc. Of ICDM, pp. 237 – 243, 2006 [10] C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, “The minimum covering energy of a graph,” Kragujevac J. Sci., vol. 34, pp. 39 – 56, 2012

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