Spectra of some Operations on Graphs

5654 | P a g e J a n u a r y 0 8 , 2 0 1 6  Spectra of some Operations on Graphs Essam El Seidy Salah ElDin Hussein Atef AboElkher 1,2. Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt 3. Department of Mathematics, Faculty of Science, Asyut University, Asyut, Egypt atefmohamed55@yahoo.com ABSTRACT: In this paper, we consider a finite undirected and connected simple graph G(E, V) with vertex set V(G) and edge set E(G).We introduced a new computes the spectra of some operations on simple graphs [union of disjoint graphs, join of graphs, Cartesian product of graphs, strong Cartesian product of graphs, direct product of graphs].


INTRODUCTION
Graph theory and its applications has a long history, in structural mechanics and in particular nodal ordering and graph partitioning are well documented in the literature, Kaveh [11][12]. Algebraic graph theory can be considered as a branch of graph theory, where eigenvalues and eigenvectors of certain matrices are employed to deduce the principal properties of a graph. In fact eigenvalues are closely related to most of the invariants of a graph, linking one extremal property to another. These eigenvalues play a central role in our fundamental understanding of graphs. Most of the definitions on algebraic graph theory in the present interesting books such as Biggs [2], Cvetković et al. [5], and Godsil and Royle [10]. One of the major contributions in algebraic graph theory is due to Fiedler [9], where the properties of the second eigenvalue and eigenvector of the Laplacian of a graph have been introduced. This eigenvector, known as the Fiedler vector is used in graph nodal ordering and bipartition, Refs. [14][15][16][17].
The Laplacian matrix of a graph and its eigenvalues can be used in several areas of mathematical research and have a physical interpretation in various physical and chemical theories. The related matrix -the adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. In the same time, the Laplacian spectrum is much more natural and more important than the adjacency matrix spectrum because of it numerous application in mathematical physics, chemistry and financial mathematics (see papers [1,3,4,6,7,8]).
•The adjacency matrix, A = A G = (a ij ) of Gis an n × n symmetric matrix,G(finite undirected and connected simple graph) otherwise. .
•The Laplacian matrix of G is the matrix L = L G = l ij = D − A, Where D is a diagonal degree matrix D = d 1 , d 2 , … , d n of a graph G, is the degree of vertex i.
•The characteristic polynomial of is defined as , = -.
• The roots of P( , ) are the eigenvalues of ( ). We will call them also the eigenvalues of .
•The (ordinary) spectrum of a finite graph is by definition the spectrum of theadjacency matrix ( ), that is, its set of eigenvalues together with their multiplicities.

Some operations on graphs and spectra
In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. For the other operations, we assume that and are graphs with disjoint vertex-sets, ( ) = { 1 , 2 , . . . , } and ( ) = { 1 , 2 , . . . , }: (1) The union ∪ has vertex-set ( ) ∪ ( ) and edge-set ( ) ∪ ( ).
(2) The join + is obtained from ∪ by adding all of the edges from vertices in to those in .   Proof. For any square matrices 1 , 2 , 3 , … not necessarily of the same order.The claim follows at once from the relation J a n u a r y 0 8 , 2 0 1 6 Theorem Kel'mans [13] 2.2, Let + denote the join of and , then Where 1 and 2 are orders of and , respectively , is the characteristic polynomial of the Laplacian matrix of .
•The wheel graph, +1 the graph which is given by +1 = 1 + , where is the cycle graph with n vertices and 1 is any new vertex.
•The fan graph, +1 the graph which is given by +1 = 1 + , where is the path graph with n vertices and 1 is any new vertex.   (2) Find Laplacian spectrum of the complete bipartite graph K m,n .
Solution: (1) We want to show that K n , = ( − ) −1 . To solve this problem we use induction by number of vertices . For, K 1 is a singular vertex. Its Laplacian matrix Hence the statement is true for = 1. Suppose that for given the equality K n , = ( − ) −1 is already proved. It is easy to see that K n+1 is a join of K n and K 1 . By Kel'mans theorem we get Hence, the Laplacian spectrum of K n = 0, n n−1 .

Types of graph products and spectra
(1) Cartesian product × □ has the vertex-set ( ) × ( ), and , is adjacent to ( ℎ , ) if either: (a) is adjacent to ℎ in and = ,or (b) = ℎ and is adjacent to in .     (ii) The adjacency matrix of × can be written as ( ) ⊗ + ⊗ ( ).
Here ⊗ is tensor (kronecker) product of matrices.
(iii) The adjacency matrix of ⊗ can be written as ⊗ .
(iv) The adjacency matrix of ⊠ can be written as + ⊗ + − .
(v) The Kronecker product of two matrices A and B, is the matrix we get by replacing the ij − th entry of A by aijB, for all i and j.
Proof. (1)Let and be the adjacency matrix of and respectively. For any eigenvalue and eigenvector of , any eigenvalue µ and eigenvector of . We have = and = µ . It follows that Thus, + µ is an eigenvalue of × . (2) Thus, µ is an eigenvalue of ⊗ . Thus, + 1 µ + 1 − 1is an eigenvalue of ⊠ . □ J a n u a r y 0 8 , 2 0 1 6

CONCULSION
In this paper, we give a new computes of laplacian spectrum of some graphs which represent addition of two graphs. Also we give a new computes of adjacency spectrum of some graphs which represent product of two graphs.