International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518
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On Spectral Radius of Graphs ππππππππ πΊ, π
π
π
π
π
π
πΎπΎπΎπΎπΎ π. π
, πππππ πΎπΎπΎπΎπΎ V.
AbstractβLet G(V, E) be simple graph with n vertices and m edges and A be vetex subset of V(G ). For any vβ π΄ the degree of the vertex π£π with respect to the subset A is defined as the number of vertices A that are adjacent to π£π . We call it as π βdegree and is denoted by ππ . Denote π1 (πΊ) as the largest eigenvalue of the graph G and π π .as the sum of π degree of vertices that are adjacent to π£π . In this paper we give lower bounds of π1 (πΊ) in terms of π degree. Index Termsβπ-degree, spectral radius, dominating set, covering set, adjacency matrix.
ββββββββββο΅ββββββββββ
1 INTRODUCTION
L
et G be simple graph with n vertices and m edges. For any π£π β π, the degree of π£π , denoted by ππ , is the number of edges that are adjacent to π£π . 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=β
then ππ =0 and if A=V then ππ = d(π£π ) β π£π β π.
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BOUNDS FOR SPECTRAL RADIUS
Theorem 2.1. Let G be a connected graph with n vertices as π£1 , π£2 , β¦ , π£π . If π1 , π2 , . . . , ππ represents π-degree sequence of these vertices with respect to a vertex subset A of G then
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For any graph G, let A(G)=(πππ ) be the adjacency matrix. The eigenvalues π1 , π2 , β¦ , ππ of A assumed in non-increasing order. The largest eigenvalue π1 (πΊ) is called spectral radius of the graph. Till now, there are many bounds for π1 (πΊ ) (see for instance [3,6]). Further results and properties see the reviews [1,2,4,5]. In this paper, we give bounds for π1 of G in tems of π-degree, which is generalization of earlier bounds found in the paper [3,6]. Using minimum dominating set and minimum covering set, the adjacency matrices π΄π· (πΊ) , π΄πΆ (πΊ) have been defined in the papers [7,10]. The π-degree is also used to get bounds for the largest eigenvalue ππ·1 (πΊ), π1πΆ (πΊ) of these matrices. We start with defining degree of vertex with respect to a subset of vertex set or π-degree. Definition1.1: Let G(V, E) be a simple graph and A be any vertex sub-set. The degree of a vertex π£π of a graph G with respect to A is the number of vertices of A that are adjacent to π£π . This degree is denoted by ππ΄ (π£π ) ππ ππ .
π1 (πΊ) β₯ οΏ½
Where, s i is sum of π degree vertices that are adjacent to v i . Equality case occurs when A=V
Proof. Let X = (π₯1 , π₯2 , β¦ , π₯π )π be a unit positive eigenvector of A(G) corresponding to π1 (G). Let C = οΏ½βπ
β’ ππππππππ πΊ, π
π
π
π
π
π
πΎπΎπΎπΎπΎ π. π
, Post Graduate Department of Mathematics, Maharaniβs Science College for Women, J. L. B Road, Mysore-570005, India. E-mail :
[email protected],
[email protected] β’ πππππ πΎπΎπΎπΎπΎ V. Assistant Professor of Physics, Government College for Women, Hole Narasipura, Hassan, Indias E-mail :
[email protected]
1
2 π=1 ππ
AC = οΏ½βπ
1
2 π=1 ππ
= οΏ½βπ
1
2 π=1 ππ
(π1 , π2 , β¦ , ππ )π
(βππ=1 π1π ππ , β¦ , βππ=1 πππ ππ )π (π 1 , π 2 , β¦ , π π )π
Similarly πΆ π π΄ = οΏ½βπ 1
ββββββββββββββββ
π 12 + π 22 + β¦ + π π2 π12 + π22 + β¦ + π2π
πΆ π π΄2 πΆ= βπ
2 1 ππ
1
2 π=1 ππ
(π 1 , π 2 , β¦ , π π )
(π 12 + π 22 + β― + π π2 )
β΄ π1 (πΊ)= οΏ½π1 (π΄2 ) = βπ π π΄2 π β₯ βπΆ π π΄2 πΆ Hence π1 (πΊ) β₯ οΏ½
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2 π 12 +π 22 +β¦+π π
2 2 π2 1 +π2+β¦+ππ
International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518
Corollary 2.1. When A= V, π1 (πΊ) β₯ οΏ½
2 π‘12 +π‘22 +β¦+π‘π
2 2 π2 1 +π2 +β¦+ππ
where π‘π represents the sum of degree of vertices that are adjacent to π£π . Theorem 2.2. Let G be a connected graph with π-degree sequence of vertices as π1 , π2 , π3 , β¦ , ππ corresponding to the subset A of G then
(π1 π1 + π2 π2 + β― + ππ ππ )2 π1 (πΊ) β₯ οΏ½ π(π12 + π22 + β¦ + π2π )
Proof. By theorem2.1, π1 (πΊ ) β₯ οΏ½
2 π 12+π 22 +β¦+π π
By Cauchyβs Schwarz inequality,βππ=1 π π2 β₯ β΄π1 (πΊ) β₯ οΏ½
2 (βπ π π π)
2 2 π(π2 1 +π2 +β¦+ππ )
Also π 1 + π 2 + β― + π π = π1 π1 + π2 π2 + β― + ππ ππ
βπ π=1 ππ π
Let C = οΏ½βπ
(π1 π1 + π2 π2 + β― + ππ ππ )2 π1 (πΊ) β₯ οΏ½ π(π12 + π22 + β¦ + π2π )
Since ππ β₯ ππ , π1 (πΊ ) β₯ οΏ½
2 βπ π=1 ππ
(D+A)C=οΏ½βπ = οΏ½βπ
π2
π1 (πΊ ) β₯
1
2 π=1 ππ
Similarly , 2 (βπ π ππ )
(π1+π2 +β―+ππ )2
where k = |D|
(π1 , π2 , β¦ , ππ )π
(βππ=1(π1π + π1π )ππ , β¦ , βππ=1(πππ + πππ )ππ )π
(π 1 + βππ=1 π1π ππ , β¦ , π π + βππ=1 πππ ππ )π π
π
π=1
π=1
1 πΆ π (π· + π΄) = οΏ½ π οΏ½π 1 + οΏ½ πππ ππ , β¦ , π π + οΏ½ πππ ππ οΏ½ βπ=1 π2π
π
βππ=1 ππ
Therefore πΆ π (π· + π΄)2 πΆ =
π
But βππ=1 ππ = βππ=1 ππ where k = |A| π1 (πΊ ) β₯ (πβ1) π
.
π
2
π
2
1 π 2 οΏ½οΏ½π 1 + οΏ½ πππ ππ οΏ½ + β― + οΏ½π π + οΏ½ πππ ππ οΏ½ οΏ½ β1 ππ
βπ π=1 ππ
Corollary 2.3. If A = D , a minimum dominating set, Then βππ=1 ππ β₯ (π β 1) and so π1 (πΊ ) β₯
1
2 π=1 ππ
(π21 +π22 +β―+π2π ) π
1
2 π=1 ππ
(π21 +π22 +β―+π2π )2 π(π21+π22+β¦+π2π)
By Cauchyβs Scwarz inequality, βππ=1 π2π β₯ Thus π1 (πΊ ) β₯ οΏ½
2 π 2 βπ π=1 π π + βπ=1 ππ
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subset A with |A|= k then π1 (πΊ ) β₯
β₯οΏ½
1 ππ π = π πππ π£π β π· 0 ππβππππππ
Where, s i is sum of π degree vertices that are adjacent to v i . Equality case occurs when D =V. Proof. Let X = (π₯1 , π₯2 , β¦ , π₯π )π be a unit positive eigenvector of π΄π· (πΊ) corresponding to ππ· 1 (πΊ)).
Corollary 2.2. For a connected graph G and a vertex
Which implies π1 (πΊ )
Definition 2.1. The minimum dominating adjacency matrix π΄π· (πΊ)is defined by π΄π· (πΊ) = π·(πΊ)+ A(G) , where A(G) is the adjacency matrix and D(G ) =(πππ ) is π Γ π matrix with
ππ· 1 (πΊ) β₯ οΏ½
(π1 π1 + π2 π2 + β― + ππ ππ )2 π1 (πΊ) β₯ οΏ½ π(π12 + π22 + β¦ + π2π )
Proof: By theorem 2.2
We recall the definition of minimum dominating matrix π΄π· (πΊ), as defined in [7].
Theorem 2. 3. Let G be a connected graph with n vertices and D be a minimum dominating set. If π1 , π2 , . . . , ππ represents π-degree sequence of these vertices corresponding to the vertex subset D of G and π΄π· (πΊ) is minimum dominating matrix then
π
(π 1 +π 2 +β―+π π )2
Corollary 2.4. If A=V then βππ=1 ππ = βππ=1 ππ = 2π 2π therefore, π1 (πΊ ) β₯ π .
πππ =οΏ½
2 2 π2 1 +π2 +β¦+ππ
2
π
π=1
β₯
1
2 βπ 1 ππ
2
[π 12 + π 22 β¦ + π π2 +(βππ=1 π1π ππ )2 + β― + οΏ½βππ=1 πππ ππ οΏ½ ]
Hence πΆ π (π· + π΄)2 πΆ β₯
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π=1
1
2 βπ 1 ππ
[π 12 + π 22 β¦ + π π2 + βππ=1 π2π ]
International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN 2229-5518
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where k = |D|
β΄ π1π· (πΊ)= οΏ½π1 (π· + π΄)2 = οΏ½π π (π· + π΄)2 π β₯ οΏ½πΆ π (π· + π΄)2 πΆ
β΄
π1π· (πΊ)
π
π βπ π 2 + βπ=1 π2π β₯ οΏ½ π=1 π π 2 βπ=1 ππ
2 π (π1 π1 +π2 π2 +β―+ππ ππ + βπ=1 ππ )
ππ· 1 (πΊ) β₯ οΏ½
οΏ½οΏ½ π2π π=1
2 βπ π=1 ππ
οΏ½π βππ=1 π2π
π (βπ π=1 ππ + βπ=1 ππ )
π
βππ=1(ππ + ππ ) π
π1π· (πΊ) β₯
π1π· (πΊ) β₯
2π + π π
Proof. Since ππ β₯ ππ , we get new bounds for π1π· (πΊ) (βππ=1 π2π + βππ=1 π2π ) οΏ½π βππ=1 π2π π+π
But for any real numbers a and b
(βππ=1 π2π + βππ=1 π2π )
β₯
βπ
β₯ οΏ½(π + π) so
π
οΏ½(οΏ½ π2π π=1
+
π
π
π
π=1
π=1
οΏ½ ππ = οΏ½ ππ β₯ π
Therefore ππ· 1 (πΊ ) β₯
2π+π π
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
π π2π οΏ½βπ=1
ππ· 1 (πΊ ) β₯
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π (βππ=1 π π2 + βπ=1 π2π )2 π 2 π βπ=1 ππ
2 (π1 π1 + π2 π2 + β― + ππ ππ + βπ π=1 ππ )
ππ·1 (πΊ) β₯
π=1
2
When D = V, βππ=1 ππ = βππ=1 ππ =2m and
2 π 2 βπ π=1 π π + βπ=1 ππ
Also π 1 + π 2 + β― + π π = π1 π1 + π2 π2 + β― + ππ ππ
ππ· 1 (πΊ) β₯
(βππ=1 ππ + βππ=1 ππ ) π
+ οΏ½ π2π οΏ½ β₯
Since βππ=1 ππ = βππ=1 ππ where k = |D|
By Cauchy Schwarz inequality, the above relation simplifies to π1π· (πΊ) β₯ οΏ½
π
The above relations simplifies to
where k = |D|
οΏ½π βππ=1 π2π
Proof. By theorem 2.3
βπ
Using Cauchy Schawarz inequality
Theorem 2. 4. Let G be a connected graph with n vertices and D be minimum dominating set. If π1 , π2 , . . . , ππ represents π-degree sequence of these vertices corresponding to the vertex subset D of G and π΄π· (πΊ)is minimum dominating matrix then
ππ· 1 (πΊ) β₯
οΏ½(βππ=1 π2π + βππ=1 π2π )
ππ· 1 (πΊ) β₯
Theorem2.6. Let G be a connected graph and C be a minimum covering set. The minimum adjacency covering matrix, π΄πΆ (πΊ) defined by
π΄πΆ (πΊ) = π·(πΊ)+ A(G) , where A(G) is the adjacency matrix and D(G ) =(πππ ) is π Γ π matrix with 1 ππ π = π πππ π£π β πΆ πππ =οΏ½ 0 ππβππππππ
Then ππΆ 1 (πΊ) β₯ οΏ½ π1πΆ (πΊ) β₯
2 π 2 βπ π=1 π π + βπ=1 ππ 2 βπ π=1 ππ
(π1 π1 +π2 π2 +β―+ππππ+ βππ=1 π2 π) 2 οΏ½π βπ π=1 ππ
π1π· (πΊ) β₯
where, c is covering number
οΏ½ π2π ) π=1
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where c = |C|
2π + π π
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 π-degree. It was proved that these bounds are generalization of earlier bounds found in the papers[5,6].
REFERENCES D. Cvetkovic, I. Gutman, eds, βApplications of Graph Spectra,β 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, βBounds for the eigenvalues of a graph, Graphs, Combinatorics, algorithms and applications,β Narosa publishing House, New Delhi, 2005 [4] D. Cvetkovic, M. Doob, H. Sachs, βSpectra of graphs β 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, βOn Spectral radius of graphs,β Linear algebra and its application, vol. 387, pp. 41-49, 2004 [7] M. R. Rajesh Kanna, B. N. Dharamendra, G. Sridhara, β Minimum dominating energy of a graphs,β International Journal of Pure and Applied Mathematics , vol. 85(4), pp. 707 - 718, 2013 [8] M. R. Rajesh Kanna, B. N. Dharamendra, G. Sridhara, β Some results on the degree of a vertex of graph with respect to any vertexset,β International Journal of Contemporary Mathematical Sciences, vol. 8, pp. 1-4, 2013 [9] S. S. Kamath and R. S. Bhat, βSome new degree concepts in graphsββ Proc. Of ICDM, pp. 237 β 243, 2006 [10] C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, βThe minimum covering energy of a graph,β Kragujevac J. Sci., vol. 34, pp. 39 β 56, 2012
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