EQUITABLE COLORING BY SPECTRAL CLUSTERING IN THE DISTRIBUTED WEB NETWORKS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The limited number of servers and more number of users in a distributed system is a great concern in load balancing. In this paper the efficient way of allocating tasks to available servers using equitable coloring of graphs is studied. For balancing the load, the complexity of graph is reduced by spectral clustering, which partitions the graph for further processes. The web model of graphs such as generalized structure of wheel and web graphs are investigated. The spectrum of these graphs are calculated for partitioning and then equitable coloring is applied. This approach results in the proper distribution of load in the web networks.


INTRODUCTION
The graph considered in this paper is an undirected web structure graphs namely generalized wheel graph and generalized web graph. Both these graphs starts from the hub, which imitates the connections of clients to the server. The clients are taken as the vertices at different layers of the cycles in these graphs. Likewise the distributed network may have few servers providing service to more number of clients in a complex network. The aim of load balancing is the equitable distribution of work among all the reliable servers in the network. This can be acheived by grouping and allocating users equally in the hotspot among the servers.
Our focus is to design and implement the load balancing with equitable coloring by spectral partitoning and then assigning colors. The spectrum is obtained by the eigenvalues of the adjacency matrix of W (n,m) and W B (n,m) . For partitioning the eigenvector of the λ th mn eigenvalue is calibrated and the vertices are clustered according to the positive and negative elements of the eigenvector X mn . Further the cut edge disconnects the clusters from the central point and adjacent layers of cycles. Later the coloring of the vertices is done based on the adjacency within the clusters, which satisfies the condition of equitablility. In 1973, Meyer [1] introduced the idea of equity in vertex coloring, this concept helps in balancing the load in a distributed system.
The graph spectra appears to be one of the interesting concept in computer science and network analysis, which helps in the partition of graphs using eigenvalues. The study on spectra of graphs can be found in many papers, see e.g. [2,3]. The proposed method of spectral clustering is a new approach to reduce the complexity of graph. The idea of applying equitable coloring in load balancing technique is a novel perspective. In this paper, the spectral clustering and optimal coloring in an equitable way for generalized W (n,m) and W B (n,m) are studied and illustrated.   Definition 2.5. [7] A generalized web graph W B (n,m) , n ≥ 3, m ≥ 2, is a graph obtained by joining all vertices v (i,m) , 1 ≤ i ≤ n, of the generalized prism P m n to a further vertex w, called the centre. Thus W B (n,m) contains mn + 1 vertices and 2mn edges.
Definition 2.7. [9] A cut vertex of a graph G is a vertex whose removal increases the number of connected components of the graph. A cut edge of a graph is defined in a similar manner as an edge whose removal increases the number of connected components of the graph. For a connected graph of order n has atmost n − 2 cut vertices and atmost n − 1 cut edges.
Definition 2.8. [10] If the set of vertices of a graph G can be partitioned into k classes V 1 ,V 2 , ...,V k such that each V i is an independent set and the condition ||Vi| − |V j|| ≤ 1 holds for every pair (i, j), then G is said to be equitably k-colorable. The smallest integer k for which G is equitably k-colorable is known as the equitable chromatic number of G and denoted by χ = (G).

EQUITABLE COLORING ON SPECTRAL CLUSTERING OF GENERALIZED WHEEL AND WEB GRAPHS
Theorem 3.1. The equitable coloring of n-wheel graph W (n,m) with n partitioned clusters is Proof. The n-wheel graph consists of mn + 1 vertices and 2mn edges. Its adjacency matrix is formed by 1, i f i and j are adjacent 0, i f i and j are non-adjacent.
The adjacency situations of n-wheel graph are with 1's on The non-adjacency situations are with 0's on i = j and elsewhere. Here n represents the number of cycles and m is the number of vertices on each cycle. Therefore the adjacency matrix is constructed as follows.
The clusters are For the vertex v 1 in the cluster C 1 , assign the color 1 in both the following cases.
Case 1. If m is odd, the vertices of the remaining clusters can be grouped as (i)If m ≡ 0 mod 3 and m ≡ 2 mod 3, the colors assigned for its vertices are

Case 2. If m is even
For the vertices v k (2 ≤ k ≤ mn + 1) in the clusters C 2 ,C 3 , . . . ,C n+1 , the colors are assigned as As the graph is sub-divided, note that the coloring is done based on the adjacency within the clusters such that the equitable condition ||V i | − |V j || ≤ 1, i = j, is attained. It is clear that 2, i f m is even.

Illustration:
Consider the generalized wheel graph of order W (4,4) with 4 cycles and having 4 vertices on each cycle. The eigen values of the adjacency matrix for this graph are The eigenvector X corresponding to λ 16 is , v 17 }, afterwards the coloring is allocated and found to be equitable as shown in figure 1.  The adjacency matrix of generalized web graph is with 1's on i = 1, j = 2, 3, . . . , m + 1, The non-adjacency situations are with 0's on i = j and elsewhere. Here n represents the number of cycles and m is the number of vertices on each cycle.
By finding the characteristic equation of this adjacency matrix and determining its eigenvalues and eigenvectors, the vertices are clustered, subsequently followed by equitable coloring.
The remaining proof is similar to theorem 3.1.

Illustration:
Consider the generalized web graph of order W B (4,7) with 4 cycles and having 7 vertices on each cycle. The eigenvalues of the adjacency matrix for this graph are

CONCLUSION
The spectral clustering has extensive applications in the field of chemical compounds of molecular graphs and protein structures etc. It is hard to cluster the the natural complex structure of graphs. By relating the spectrum, the graph is partitioned and graph coloring is applied which enables the idea of sharing of resources to the craving users. This paper suggests the method of spectral partitioning and equitable coloring of graphs. The model of generalized graphs W (n,m) and W B (n,m) are investigated and illustrated. Moreover this procedure can be executed for various model of complex structured graphs.

ACKNOWLEDGEMENT
With due respect, the author sincerely thanks the Karunya Institute of Technology and Sciences, Coimbatore, India, for providing financial assistance through KSTG seed money. Also express his gratitude to the reviewers for their wise suggestions.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.