3.

SPECTRA OF VERTEX DISTANCE COMPLEMENT MATRIX(VDC)The eigenvalue of VDC(G)is called the VDC – eigenvalue of the graph G. It is denoted by  ?1? ?2 ? … ? ?n. The set of all VDC – eigenvalues of G is called the VDC- spectrum of G and isdenoted by VDCSpec(G). Two non isomorphic graphs aresaid to be VDC – cospectral if they have the same VDC – spectrum.Theorem 3.

1. Let G be a r – regular graph on ‘n’vertices and diam(G) = 2. Let { r = ?1, ?2, …, ?n}be the adjacency eigenvalues ofG then VDC – eigenvalues of G are (n – 1)(n – 2) + r and             ?i  – n + 2 for i = 2, 3, … ,  n.Proof. Given G is r – regular. We have ris the largest eigenvalue of A = A(G). Let D = D(G) be thedistance matrix of the graph G.

Since diam(G) = 2,D(G)= A + 2 = A + 2(J  –  I  –  A)= 2(J  –  I) – AFrom Definition 2.1 VDC – matrix, VDC(G)can be written as,VDC(G) = n(J  –  I)   – D                                                          = (n  – 2)J   – (n- 2)I + A                                                          = (n  – 2)(J – I) + AWhere J is an n  nmatrix with all entries equal to 1 and Iis the identity matrix.Using Lemma 2.1 we get the VDC – eigenvalues are (n  – 1)(n   – 2) + r and  ?i – n + 2 for i = 2, 3, …, n.Theorem 3.

2. Let G be a connectedr – regular graph with n> 3 vertices and let none ofthe three graphs Fi, i = 1, 2, 3 of Figure.1 will be an induced subgraph ofG. Then VDC – spectrum of L(G) isVDCSpec(L(G)) = for i = 2, 3, …, n.Proof.

Let { r = ?1,?2, …, ?n } be the adjacency spectrum of G. Then by Theorem 2.3 L(G), is a(2r  – 2) – regular graph with spectrum 2r – 2,  ?i + r – 2 for i = 2, 3, …,  nand  -2 repeats times.

Also Fi, i= 1, 2, 3 is not an inducedsubgraph of G, by Theorem2.5,  diam(L(G)) ?  2.Hence by theorem 3.1 VDC – eigenvalues of L(G) are,             (– 1)( – 2) + 2r – 2 = ¼(n2r2– 6nr + 8r)                                                       (1)                       (?i + r – 2)–   + 2 = ½ (2?i+ 2r – nr)           for    i =2, 3, …, n              (2)                                   – 2 –   + 2 = –        repeats         times                               (3)Theorem 3.3. Let G be a connected r- regular graph with n > 3 vertices and for any two non adjacent vertices ‘u’ and ‘v’of a graph G, there exists a third vertex ‘w’ which is not adjacent to either ‘u’ or ‘v’.

ThenV DC – spectrum of   is, VDCSpec( =  for i = 2, 3, …, n.Proof : Let { r = ?1, ?2, …, ?n} be the adjacency spectrum ofG. Then by Theorem (2.4) , is a ( – 2r+ 1) – regular graph on  verticeswith spectrum – 2r + 1, – ?i – r + 1  for  i =2, 3, …, n and 1 repeats   times.

By Theorem3.1 the VDC – spectrum of   is,              ( – 1)( – 2)  +  – 2r +1-?i–r + 1 –  + 2         for       i= 2, 3, …, n                                         1 – + 2 = 3 –             repeats         times.ie,()2 – nr – 2r + 3 = ¼ (n2r2 – 4nr – 8r +12)                                             (4)     -½ (2?i + 2r + nr – 6),      for   i = 2, 3, …, n                                                (5)                3 – ,       repeats     times                                         (6)Theorem 3.4. Let G be a connectedr – regular graph with n> 3 vertices and let none ofthe four graphs Fi,  i = 1,  2, 3, 4 of Figure .1 willbe an induced subgraph of G. Then VDC – spectrum ofL2(G) isVDCSpec(L2(G)) =     for i = 2, 3, …, n  and k = .

Proof. Let { r = ?1,?2, …, ?n } be the adjacency spectrum of G.  Thenby Theorem 2.3 and Theorem 3.1 L2(G), is a (4r – 6) – regular graph onk =   verticeswith VDC –spectrum(k – 1)(k – 2) + 4r – 6  (?i + 3r – 6) –k + 2,    for  i=2, 3, …, n2r – 6 – k + 2,    repeats      times     – 2 – k + 2,     repeats     timesie                                 k2- 3k + 4r – 4,      where  k =                                        (7)  ?i + 3r – k – 4,        for i=2, 3, …, n                                    (8)2r – k – 4,      repeats      times                              (9)-k,      repeats      times                             (10)

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