1 / 17

GRAPHS AND MATRACIS- AN ANALYSIS

GRAPHS AND MATRACIS- AN ANALYSIS. Dr.B.Basavanagoud Department of Mathematics Karnatak University Dharwad-580 003. The Theory of graphs properly be classified as a subfield of Matrix Theory. Adjacency Matrix A(G) of a graph G with V(G) = A(G) = where

ulric
Download Presentation

GRAPHS AND MATRACIS- AN ANALYSIS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. GRAPHS AND MATRACIS- AN ANALYSIS Dr.B.Basavanagoud Department of Mathematics Karnatak University Dharwad-580 003

  2. The Theory of graphs properly be classified as a subfield of Matrix Theory. • Adjacency Matrix A(G) of a graph G with V(G) = A(G) = where v1 v2 0 1 0 1 0 1 0 0 1 1 G: v3 A(G): 0 0 0 0 1 1 1 0 0 1 v5 v4 0 1 1 1 0 A(G) is (i): (0,1) matrix (ii): symmetric matrix (iii): zero diagonal

  3. v1 v2 v3 v4 v5 v1 v1 0 1 1 1 0 v2 1 0 0 1 0 v2 v3 A(G)=v3 1 0 0 1 1 G: v4 1 1 1 0 1 v5 0 0 1 1 0 v4 v5 The highest powers of A has analogous property.

  4. Powers of Adjacency Matrix: 2 1 0 1 0 2 5 2 5 2 1 3 1 2 1 5 4 1 5 6 0 1 1 1 0 2 1 0 1 3 1 2 1 3 1 5 5 1 4 6 2 1 0 1 3 2 6 3 6 2

  5. Distance in Graphs • For a connected graph G, we define the distance d(u,v) between two vertices u and v as the length of any shortest u-v path. If there is no path connecting u and v, we define d(u,v) =∞. The distance matrix D=[ dij] of a connected graph G of order p with V(G)= {v1, v2,….,vp} that p-by –p matrix for which dij is the distance between vi and vj.

  6. b a c e dThe distance matrix is therefore a symmetric matrix with non-negative integer entries having zero diagonal.The following theorem characterizes those matrices which are the distance matrix of some graph.

  7. THEOREM1. A p by p matrix D= [dij] is the distance matrix of a graph of order p if and only if D has the following properties:(i) dij is non-negative integer for all i,j(ii) dij =0 if and only if i=j(iii) D is symmetric (iv) dij ≤ dik+dkj for all i,j,k and (v) For dij>1, there exists k≠ i,j , such that dij = dik+dkj. There is an interesting class of graphs which one can associate with a given graph G of order p based on the distance concept. These are the powers of G.The nth power Gn of G is that graph with V(Gn)=V(G) for which uv∊ E(Gn) if and only if 1≤ d(u,v)≤ n in G.

  8. The graphs G2 and G3 are also referred to as the square and cube, respectively of G. G: G2: G3:

  9. The square of G, denoted by G2, has the points of G and the points u and v adjacent in G2 if and only if they are joined in G by a path of length 1 or 2.This concept was introduced by Harary and Ross. A criterion for a given graph to be the square of some graph was found by Mukhopadhay. Let A be the adjacency matrix of G, and let I be the p-by-p identity matrix. If we compute (A+I)n-I using boolean arithmetic (1+1=1), then we arrive at a matrix which is the adjacency matrix of some graph. This graph is Gn.

  10. This observation is a direct consequence of a fact that employing boolean arithmetic, (A+I)n-I =An+An-1+……+A, and this matrix has (i,j) entry 1 of there is a path of length k, 1≤ k ≤ n, between vi and vj and has (i, j) entry 0 otherwise. A graph with its square and cube are shown in the previous slide. v1 v2 v3 v4 v1 v2 v3 v4 v1 0 1 2 3 v1 1 0 0 0 v2 1 0 1 2 v2 0 1 0 0 A= v3 2 1 0 1 I= v3 0 0 1 0 v4 3 2 1 0 v4 0 0 0 1

  11. Definition: A graph H is an nth root of G if Hn=G.The square roots of K4 are shown in the following figure: G1: G2: G3: G4: G5: The square roots of K4

  12. For Square roots, however, a criterion has been obtained.Theorem: The connected graph G of order p with V(G)={v1,v2,….,vp} has a square root if and only if G contains a collection of complete subgraphs G1,G2,….,Gp such that(i) UE(Gi)= E(G)(ii) Gi contains vi and (iii) Gi contains vj if and only if Gj contains vi.

  13. The adjacency matrix A(G) of a graph G with vertex set V(G)= {v1,v2,…,vp} is the p-by-p matrix [aij] where aij=1 if vivj∊E(G) and aij=0 otherwise. The following Figure shows a labeled graph and its adjacency matrix. v1 v2 v3 v4 v1 0 1 0 0 G: v2 1 0 1 1 A(G) = v3 0 1 0 1 v4 0 1 1 0

  14. It is evident that A(G) is a (0,1) symmetric matrix with zero diagonal.A(0,1) matrix is a matrix each of whose entries is 0 or 1.Likewise, it is clear that these conditions are sufficient for a matrix to be the adjacency matrix of some graph.Thus, the set of all such matrices for all positive integers p represents a class of all graphs. The entries of nth power An of A have particularly a nice interpretation.

  15. 1 0 1 1 0 3 1 1 0 3 1 1 3 2 4 4 A2: 1 1 2 1 A3: 1 4 2 2 1 1 1 2 1 4 3 2 Theorem: If A is the adjacency matrix of a graph G with V(G)= {v1,v2,……,vp}, then (i,j) entry of An, n≥1 is the number of different vi-vj walks of length n in G.The preceding Theorem has some immediate consequences.Definition: The trace tr(M) of a square matrix M is the sum of the diagonal entries of M.

  16. Corollary: If An = [aij(n)]is the nth power of the adjacency matrix A(G) of a graph G with V(G)={v1,v2,….,vp} then(i) aij(2), i≠j, is the number of vi-vj paths of length two(ii) aii(2)= deg(vi) and(iii) 1/6 tr(A3) is the number of triangles of G.

  17. THANK YOU

More Related