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E. A (W ) ⊂ W and AT (W ) ⊂ W . If {x1 , x2 ,…, xk } is an orthonormal basis for W then there 25 An Introduction to Relevant Graph Theory and Matrix Theory k exists unique real k × k matrices B and C such that if x = ∑ α i x i , i =1 k Ax = ∑ β i x i , i =1 k and A x = ∑ χi xi , T i =1 then  β1   α1   ⋮  = B ⋮       β k  α k  and  χ1   α1   ⋮  = C  ⋮  . It is easy to see that BT = C so that if A is symmetric      χ k  α k  and A (W ) ⊂ W , then AT (W ) ⊂ W and B is symmetric.

6(b). To do this, we first require a result for the eigenvalues of the adjacency matrix of the product of two graphs based on the eigenvalues of the individual graphs. , label of ui , v j ) is smaller than that of ( u , v ) if k ℓ and only if i < k or i = k , j < ℓ,) , then A(G1 × G 2 ) = A ( G1 ) ⊕ A ( G2 ) . e. 52 Spanning Tree Results for Graphs and Multigraphs  B11 B 21 A ( G1 ) ⊕ A ( G2 ) =   ⋮   Bn1 B12 ⋯ B1n  B22 ⋯ B2 n  . ⋮ ⋱ ⋮   Bn 2 ⋯ Bnn  Since the χχ block of I n ⊗ A ( G2 ) is A ( G2 ) and the χχ block of A ( G1 ) ⊗ I m is ( A ( G1 ) ) ⋅ I m = 0 , it follows that B χχ = A ( G2 ) .

C) Follows trivially from the fact H = D − A . (d) Since H1 = 0 by (c) we have that 0 is an eigenvalue, and H is singular. (e) and (f) follow trivially from the fact H = D − A . (g) Follows from the fact that H = D − A and the First Theorem of □ Multigraph Theory. 2 (a), (b) and (e) we get the eigenvalues of H ( M ) can be ordered 0 = λ1 ( H ( M ) ) ≤ λ2 ( H ( M ) ) ≤ … ≤ λn ( H ( M ) ) . Unless otherwise stated, we assume that eigenvalues are the eigenvalues of the Laplacian H ( M ) . In order to simplify notation we will use λi ( M ) or simply λi when the multigraph is understood, in place of λi ( H ( M ) ) .

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2-3 graphs which have Vizings adjacency property by Winter P. A.

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