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 is
denoted by VDCSpec(G). Two non isomorphic graphs are
said to be VDC – cospectral if they have the same VDC – spectrum.

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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 of
G 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 r
is the largest eigenvalue of A = A(G). Let D = D(G) be the
distance matrix of the graph G.
Since diam(G) = 2,

D(G)
= A + 2

= A + 2(J  –  I  –  A)

= 2(J  –  I) 
– A

From 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) + A

Where J is an n  n
matrix with all entries equal to 1 and I
is 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 connected
r – regular graph with n
> 3 vertices and let none of
the three graphs Fi,
 i = 1, 2, 3 of Figure.1 will be an induced subgraph of
G. Then VDC – spectrum of L(G) is

VDCSpec(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, …,  n
and  -2 repeats

times.

Also Fi, i
= 1, 2, 3 is not an induced
subgraph of G, by Theorem
2.5,  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’. Then
V DC – spectrum of   is,

VDCSpec( = 

for i = 2, 3, …, n.

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

By Theorem
3.1 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 connected
r – regular graph with n
> 3 vertices and let none of
the four graphs Fi,  i = 1,  2, 3, 4 of Figure .1 will
be an induced subgraph of G. Then VDC – spectrum of

L2(G) is

VDCSpec(L2(G)) = 

    for i = 2, 3, …, n  and k = .

Proof. Let { r = ?1,
?2, …, ?n } be the adjacency spectrum of G.  Then
by Theorem 2.3 and Theorem 3.1 L2(G), is a (4r – 6) – regular graph on
k =   vertices
with VDC –spectrum

(k – 1)(k – 2) + 4r – 6

  (?i + 3r – 6) –
k + 2,    for  i=
2, 3, …, n

2r – 6 – k + 2,    repeats      times

     – 2 – k + 2,     repeats     times

ie                                 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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