On Strong Vertex Graceful Graphs
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1 International Mathematical Forum, 5, 010, no 56, On Strong Vertex Graceful Graphs D D Somashekara and C R Veena Department of Studies in Mathematics, University of Mysore Manasagangothri, Mysore , India dsomashekara@yahoocom veenacrmaths@gmailcom Abstract In this paper we find the energy, the Laplacian energy and the Laplacian-energy-like invariant of an (n, n 3) strong vertex graceful graph Also we find the cardinality of the set of all (n, n 3) strong vertex graceful graphs Mathematics Subject Classification: 05C78, 05C50 Keywords: strong vertex graceful graph 1 Introduction A graph labelling is an assignment of integers to the vertices or edges or both, subject to certain conditions Graph labelling is one of the most interesting problems in Graph theory and they serve as useful models for a broad range of applications such as coding theory, X-ray crystallography, circuit designs etc Graph labellings were first introduced in the late 1960s and have been motivated by practical problems In the intervening years variety of graph labelling techniques have been studied and the subject is growing exponentially For more details one may refer the survey article [1] by J A Gallian The graceful labelling methods were introduced by Rosa [6] in 1967 A graph G with p vertices and q edges is graceful if f is an injection from the vertices of G to the set {0, 1,,p} such that, when each edge uv is assigned the label f(u) f(v), the resulting edge labels are all distinct S M Lee, Y C Pan and M C Tsai [4] have recently introduced the concept of strong vertex graceful graph by giving following definition Definition: A graph G is said to be strong vertex graceful if there exists a bijective mapping f : V (G) {1,,,n} such that for the induced labelling f + : E(G) N defined by f + ((u, v)) = f(u) +f(v), f + (E(G)) consists of
2 75 D D Somashekara and C R Veena consecutive integers Note: If we replace N by Z n, where n is the number of edges in G, we get a bijective map Lee, Pan and Tsai [4] called such graphs vertex graceful They observe that the class of vertex graceful graphs properly contains the super edge-magic graphs and strong vertex graceful graphs are super edgemagic They provided vertex graceful and strong vertex graceful labelling for various graphs of small orders For example the following graph G of order 4 and size 5 is strong vertex graceful Note: In a strong vertex graceful graph with n vertices v 1,v,,v n labelled 1,,,n, the maximum value of an edge will be n 1 (if there is an edge v n 1 v n ) and the minimum value of an edge will be 3 (if there is an edge v 1 v ) Hence the maximum number of edges in a strong vertex graceful graphs with n vertices will be n 3 The concept of energy of a graph was introduced by I Gutman [] in 1978 Let G be a graph with n vertices and m edges and let A =(a ij ) be the adjacency matrix of the graph The eigenvalues λ 1,λ,,λ n of A, assumed in non increasing order, are the eigenvalues of the graph G The set of eigenvalues of G, SpecG, is independent of labelling of the vertices of G As A is real symmetric, the eigenvalues of G are real with sum equal to zero Thus λ 1 λ λ n, λ 1 + λ + + λ n =0 The energy E(G) ofg is defined to be the sum of the absolute values of the
3 On strong vertex graceful graphs 753 eigenvalues of G That is, E(G) = λ i The Laplacian energy of a graph G has recently been defined by Gutman and B Zhou [3] Let G be a graph with n vertices and m edges The Laplacian matrix of the graph G, denoted by L =(L ij ), is a square matrix of order n whose elements are defined as δ i, if i = j L ij = 1, if i j and the vertices v i,v j are adjacent 0, if i j and the vertices v i,v j are not adjacent where δ i = degree of vertex v i Let μ 1,μ,,μ n be the Laplacian eigenvalues of G Laplacian energy LE(G) ofg is defined as LE(G) = μ i m n In 008, J Liu and B Liu [5] have introduced the Laplacian-energy-like invariant of a graph and showed that there is a great deal of analogy between the properties of E(G) and the Laplacian-energy-like invariant graph LEL(G) In fact they have first introduced the auxiliary eigenvalues ρ i,i =1,,,n of a graph G as ρ i = μ i, where μ i,i =1,,,n are Laplacian eigenvalues of the graph G Let G(n, m) be a graph with n vertices and m edges and let the Laplacian eigenvalues of G are μ 1,μ,,μ n Then the Laplacian-energy-like invariant of G is defined by LEL(G) = ρ i = μi In this paper we find the energy, the Laplacian energy and the Laplacianenergy-like invariant of an (n, n 3) strong vertex graceful graph Also we find the cardinality of the set of all (n, n 3) strong vertex graceful graphs Main Results Theorem 1 There exists a (n, n 3) graph which is strong vertex graceful Proof Let the vertices of the graph be v 1,v,,v n We label the vertices
4 754 D D Somashekara and C R Veena as follows: v i = i, 1 i n Next we connect v 1 to v k, k n and v n to v k, k n 1 Then the value of the edge v 1 v k is k + 1 for k n and that of the edge v n v k is n + k for k n 1 Thus the n-3 edges take the consecutive values 3, 4,,n 1 and the (n, n 3) graph becomes strong vertex graceful Theorem Let G be a (n, n 3) strong vertex graceful graph with vertices v 1,v,,v n labelled as v i = i, 1 i n and the edges are obtained by joining v 1 to v k, k n and v n to v k, k n 1 Then the energy of G is given by E(G) =1+ 8n 15 Proof The adjacency matrix of graph is The characteristic polynomial of the matrix is given by φ(g; λ) = λ n (n 3)λ n (n 4)λ n 3 = λ n 3 [λ 3 (n 3)λ (n 4)] Hence the eigenvalues of the adjacency matrix are λ 1 = 1+ 8n 15 Therefore, the energy is given by,λ = 1,λ 3 = 1 8n 15,λ k =0, 4 k n E SV G (G) = λ i i=0 = 1+ 8n 15 Theorem 3 Let G be a (n, n 3) strong vertex graceful graph as in Theorem Then the Laplacian energy of G is given by LE(G) = 4(n 4n +6) n
5 On strong vertex graceful graphs 755 Proof The Laplacian matrix of G is n n 1 The characteristic polynomial of the graph is φ(g; μ) = μ n (4n 6)μ n 1 + +( 1) n μn n 3 = μ(μ n 1 (4n 6)μ n + +( 1) n n n 3 ) Hence the eigenvalues of the Laplacian matrix are μ i = n for 1 i, μ i = for 3 i n 1 and μ n =0 Therefore, the Laplacian energy is given by LE(G) = μ i m n = 4(n 4n +6) n Theorem 4 Let G be a (n, n 3) strong vertex graceful graph as in Theorem Then the Laplacian-energy-like invariant of G is given by LEL(G) = ( n +(n 3) Proof The Laplacian eigenvalues of G are: μ i = n for 1 i, μ i = for 3 i n 1 and μ n =0
6 756 D D Somashekara and C R Veena Hence the Laplacian-energy-like invariant of a graph G is given by LEL(G) = = = ρ i μi n + +0 = ( n +(n 3) Theorem 5 If A denote the set of all (n, n 3) strong vertex graceful graphs then { (( n 3 A = )!)4 ( n 1 )3, n odd (( n )!)4 n, n even Proof We shall now compute the number of (n, n 3) strong vertex graceful graphs by showing how many such graphs can be constructed So, we start with n distinct points v 1,v,,v n which will later become the vertices a strong vertex graceful graph We label them as: v i = i, 1 i n We shall now show how to draw n 3 edges The edge with value 3 can be drawn in only one way by connecting the vertices v 1 and v Similarly in only one way the edge whose value is 4, the edge whose value is n 1 and the edge whose value is n can be drawn But the edge with value k, 5 k n 3 can be drawn in as many ways as there are partitions of k into two distinct parts with each part not exceeding n Therefore the cardinality of A is equal to the total number of ways the n 3 edges can be drawn Thus the problem reduces to finding the number of partitions of the k into two distinct parts with each part not exceeding n First let n be even If n = 4, then among the integers 3, 4, 5, 6, 7 only 5 has two partitions into two distinct parts, each part not exceeding 4 So, let n>4 Then among the integers 5, 6, 7,n 3 the number of partitions of each of the integers in the 4- tuple (i +3, i +4, n i, n i 1) for 1 i n 4, into two distinct parts, each part not exceeding n is i + 1 The integers n + 1 has n partitions into two distinct parts, each part not exceeding
7 On strong vertex graceful graphs 757 n Thus the total number of ways the n 3 edges can be drawn, when n is even is (( n )!)4 n When n is odd, the proof follows in the same lines Acknowledgments: The authors are thankful to the Department of Science and Technology, Government of India, New Delhi, for the financial support under the grant DST/SR/S4/MS:490/07 The authors are also thankful to Prof C Adiga and Prof H N Ramaswamy for their valuable suggestions References [1] J A Gallian, A Dynamic Survey of Graph labelling, The electronic journal of Combinatorics 16 (009), DS6 [] I Gutman, The energy of a graph, Ber Math-Satist Sekt Forschungsz Graz 103 (1978), 1- [3] I Gutman and B Zhou, Laplacian energy of a graph, Linear Algebra Appl 414 (006), 9-37 [4] S M Lee, Y C Pan and M C Tsai, On vertex graceful (p, p+1)-graphs, Congressus Numeranium, 17 (005), [5] J Liu and B Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun Math Comput Chem 59 (008), [6] A Rosa, On certain valuations of the vertices of a graph, Theory of Graph(Internat Symposium, Rome, July 1996), Gordon and Breach, N Y and Dunod Paris (1967) Received: April, 010
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