Degree Sum Polynomial Obtained by Some Graph Operators
B. Basavanagoud * and Anand P Barangi 1 Department of Mathematics,
Karnatak University, Dharwad - 580 003, Karnataka, INDIA.
*Corresponding author email: [email protected] and 1 [email protected].
(Received on: July 14, 2018) ABSTRACT
Degree sum matrix DS G ( ) of a graph G is a square matrix whose ( , ) i j th
entry is d i d j whenever i j , and zero otherwise, where d i is the degree of the
i th vertex of G . In this paper, we obtain the degree sum polynomial of some graph operations and degree some polynomial of the generalized xyz -point-line transformation graphs in terms of order and size of the graph considered for transformation.
2010 Mathematics Subject Classification: 05C07, 05C31.
Keywords: Degree sum matrix, Degree sum polynomial, Eigenvalues, Graph operators.
1. INTRODUCTION
Let G = ( , ) n m be a simple, undirected graph. Let V G ( ) and E G ( ) be the vertex set
and edge set of G respectively. The degree deg G ( ) v (or d G ( )) v of a vertex v V G ( ) is the
number of edges incident to it in G . The graph G is r-regular if and only if the degree of
each vertex in G is r. Let { , v v 1 2 ,..., } v n be the vertices of G and let d i = deg G ( ) v i . Let G ,
L(G) and S(G) of a graph G be complement, line graph and subdivision graph of a graph G
respectively. The partial complement of subdivision graph S G ( ) of a graph G whose vertex
set is V(G) E(G) where two vertices are adjacent if and only if one is a vertex of G and the
other is an edge of G non incident with it. For undefined terminology refer 6,9 .
There are several graph matrices in the literature. Adjacency matrix 6,12 is the one which is studied extensively. Distance matrix 2 , seidel matrix 5 , Laplacian matrix 13 , signless Laplacian matrix 7,14 , degree exponent matrix 16 etc., are the other examples for graph matrices.
The degree sum matrix have been introduced by H.S. Ramane et al. in 15 . Hosamani in 10 obtained bounds for degree sum energy.
The degree sum matrix of a graph G is an n n matrix DS G ( ) = [ ds ij ] , whose elements are defined as
= ,
0 .
i j
ij
d d if i j
ds otherwise
Let I be the identity matrix and J be the matrix whose all entries are equal to 1 . The degree sum polynomial of a graph G is defined as
DS ( ; ) = ( ( )).
P G det I DS G
The eigenvalues of the matrix DS G ( ) , denoted by 1 , 2 ,..., n are called the degree sum eigenvalues of G and their collection is called the degree sum spectra of G . It is easy to see that if G is an r-regular graph, then DS G ( ) = 2 rJ 2 rI . Therefore, for an r-regular graph
G of order n ,
1 DS ( ; ) = [ 2 ( 1)][ 2 ] n .
P G r n r (1.1)
In this paper, we obtain Degree sum polynomial obtained by some graph operations and some operators (The generalized xyz-point-line transformation graphs). The xyz transformation graphs were defined by Deng et al. in 8 . Basavanagoud in 3 studied the basic properties of the xyz transformation graphs by calling them generalized xyz -point-line transformation graphs and changing the notion of xyz -transformations of a graph G as T xyz ( ) G to avoid confusion between various transformations.
For a graph G = ( , ) V E , let G 0 be the graph with V G ( 0 ) = ( ) V G and with no edges, G 1 the complete graph with V G ( 1 ) = ( ) V G , G = G , and G = G .
Definition 1.1 8 Given a graph G with vertex set V G ( ) and edge set E G ( ) and three variables , , {0,1, , },
x y z the generalized xyz -point-line transformation graph T xyz ( ) G of G is the graph with vertex set V T ( xyz ( )) = ( ) G V G E G ( ) and the edge set
( xyz ( )) = (( ) ) x (( ( )) ) y ( )
E T G E G E L G E W where W = ( ) S G if z = , W = ( ) S G if
=
z , W is the graph with V W ( ) = ( ) V G E G ( ) and with no edges if z = 0 and W is
the complete bipartite graph with parts V G ( ) and E G ( ) if z = 1 .
Figure: Some generalized transformation graphs, T xyz of P 4
Since there are 64 distinct 3 permutations of {0,1, , } . Thus obtained 64 kinds of generalized xyz -point-line transformation graphs. There are 16 different graphs for each case when z = 0 , z = 1 , z = , z = . Among these 16 graphs with z = 0 are disconnected.
Thus we omit them in this paper.
The following theorems are helpful in proving our results.
Theorem 1.1 4 Let G be a graph of order n , size m and let v be the point-vertex of T xyz corresponding to a vertex v of G . Then
xy 1 ( ) d T v =
= 0, {0,1, , },
1 = 1, {0,1, , },
( ) = , {0,1, , },
1 ( ) = , {0,1, , }.
G
G
m if x y
n m if x y
m d v if x y
n m d v if x y
xy ( ) d T v =
( ) = 0, {0,1, , },
1 ( ) = 1, {0,1, , },
2 ( ) = , {0,1, , },
1 = , {0,1, , }.
G
G
G
d v if x y
n d v if x y
d v if x y
n if x y
xy ( ) d T v =
( ) = 0, {0,1, , },
1 ( ) = 1, {0,1, , },
= , {0,1, , },
1 2 ( ) = , {0,1, , }.
G
G
G
m d v if x y
n m d v if x y
m if x y
n m d v if x y
Theorem 1.2 4 Let G be a graph of order n , size m and let e be the line-vertex of T xyz corresponding to an edge e of G . Then
xy 1 ( ) d T e =
= 0, {0,1, , },
1 = 1, {0,1, , },
( ) = , {0,1, , },
1 ( ) = , {0,1, , }.
G
G
n if y x
n m if y x
n d e if y x
n m d e if y x
xy ( ) d T e =
2 = 0, {0,1, , },
1 = 1, {0,1, , },
2 ( ) = , {0,1, , },
1 ( ) = , {0,1, , }.
G
G
if y x
m if y x
d e if y x m d e if y x
xy ( ) d T e =
2 = 0, {0,1, , },
3 = 1, {0,1, , },
2 ( ) = , {0,1, , },
3 ( ) = , {0,1, , }.
G
G
n if y x
n m if y x
n d e if y x
n m d e if y x
2. DEGREE SUM POLYNOMIAL OBTAINED BY SOME GRAPH OPERATIONS In this section we obtain the degree sum polynomial of some basic graph operations.
Lemma 2.1. 16 If a b c , , and d are real numbers, then the determinant of the form
1 1 1 2
2 1 2 2
( )
( )
a I n a J n c J n n d J n n b I n bJ n
of order n 1 n 2 can be expressed in the simplified form as
1 2
1 2 1 2
1 1
( a ) n ( b ) n {[ ( n 1) ][ a ( n 1) ] b n n cd }.
The product 9 G H of graphs G and H has the vertex set V G H ( ) = ( ) V G V H ( ) and ( , )( , ) a x b y is an edge of G H if and only if [ a = b and xy E H ( ) ] or [ x = y and
( )
ab E G ]. □
Theorem 2.2. Let G be an r 1 -regular graph of order n 1 and H be an r 2 -regular graph of
order n 2 . Then P DS ( G H ; ) = ( 2( r 1 r 2 )) n n 1 2 1 [ 2( r 1 r 2 )( n n 1 2 1)].
Proof. Since the graphs G and H are regular graphs of degree r 1 and r 2 respectively.
Therefore the graph obtained by the cartesian product of G and H is a regular graph of degree r 1 r 2 with n n 1 2 vertices. Hence the result follows from Eq. (1.1). □
The composition 9 G H [ ] of graphs G and H with disjoint vertex sets V G ( ) and V H ( ) and edge sets E G ( ) and E H ( ) is the graph with vertex set V G H ( [ ]) = ( ) V G V H ( ) and ( , )( , ) a x b y is an edge of G H [ ] if and only if [ a is adjacent to b ] or [ a = b and x is adjacent to y ].
Theorem 2.3. Let G be an r 1 -regular graph of order n 1 and H be an r 2 -regular graph of order n 2 . Then
1 2 1
2 1 2 2 1 2 1 2
( [ ]; ) = ( 2( )) n n [ 2( )( 1)].
P DS G H n r r n r r n n
Proof. Since the graphs G and H are regular graphs of degree r 1 and r 2 respectively.
Therefore the graph obtained by the composition of two graphs G and H is a regular graph of degree n r 2 1 r 2 with n n 1 2 vertices. Hence the result follows from Eq. (1.1). □
The corona 9 G H of graphs G and H is a graph obtained from G and H by taking one copy of G and | ( ) | V G copies of H and then joining by an edge each vertex of the i th copy of H is named ( , ) H i with the i th vertex of G .
Theorem 2.4. Let G be an r 1 -regular graph of order n 1 and H be an r 2 -regular graph of order n 2 . Then
1 1
1 1 2
1 2 2 1 1 2 1 2 2
2
1 2 2 1 2
( ; ) = [ 2( )] [ 2( 1)] {[ 2( 1)( )][ 2( 1)( 1)]
4( )( 1) }.
n n n
P DS G H r n r n r n n n r
r n r n n
Proof. If G is an r 1 -regular graph of order n 1 and H is an r 2 -regular graph of order n 2 , then G H has two types of vertices, the n 1 vertices with degree R 1 = r 1 n 2 and the remaining n n 1 2 vertices are of degree R 2 = r 2 1 . Hence
1 1 1 1 2 1 1 2
1 2 1 2 1 2 1 2 1 2
2 ( ) ( )
( ) =
( ) 2 ( )
n n n
n n
n
R J I R R J n n
DE G H
R R J n n R J n I n
Therefore,
1 1 1 1 2 1 1 2
1 2 1 2 1 2 1 2 2 1 2
( ; ) = | ( ) |
( 2 ) 2 ( )
= ( ) ( 2 ) 2 )
DS
n n n
n n
n
P G H I DE G H
R I J R R J n n
R R J n n R I n R J n
Using Lemma 2.1, we get the required result. □
3. DEGREE SUM POLYNOMIAL OBTAINED BY SOME GRAPH OPERATORS In this section we obtain degree some polynomial of xyz Point Line transformation graphs.
Theorem 3.1. Let G be an r-regular graph of order n and size m . Then
1 1 2 2
( 01 ( ))
2 2
( ) = ( 2( 1)) ( 2 ) { [( 1)2 2( 1)]
4 ( 1)( 1) ( 1) }.
m n
DS T G
P m r n r m
r n m r m mn
Proof. The generalized xyz -point-line transformation graph T 01 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree r and the remaining m vertices are with degree m 1 by Theorem 1.1 and Theorem 1.2. Hence
01 2 ( ) ( 1)
( ( )) =
( 1) 2( 1)( )
n n n m
m n m m
r J I r m J
DS T G
r m J m J I
Therefore,
01 ( 01 ( )) ( ) = | ( ( )) |
( 2 ) 2 ( 1)
= ( 1) ( 2( 1)) 2( 1)
DS T G
n n n m
m n m m
P I DS T G
r I rJ r m J
r m J m I m J
Using Lemma 2.1, we get the required result. □
Theorem 3.2. Let G be an r-regular graph of order n and size m . Then
1 1 2
( 0 ( ))
2
( ) = ( 2 ) ( 2( 3 )) { [2 ( 1) 2( 1)( 3 2 )]
4 ( 1)( 1)( 3 2 ) ( 3 ) }.
n m
DS T G
P r m r r n m m r
r n m m r m r mn
Proof. The generalized xyz -point-line transformation graph T 0 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree r and the remaining m vertices are with degree m 3 2 r by Theorem 1.1 and Theorem 1.2. Hence
0 2 ( ) ( 3 )
( ( )) =
( 3 ) 2( 3 2 )( )
n n n m
m n m m
r J I m r J
DS T G
m r J m r J I
Therefore,
0
( 0 ( )) ( ) = | ( ( )) |
( 2 ) 2 ( 3 )
= ( 3 ) ( 2( 3 2 )) 2( 3 2 )
DS T G
n n n m
m n m m
P I DS T G
r I rJ m r J
m r J m r I m r J
Expanding the determinant, we get the required result. □
Theorem 3.3. Let G be an r regular graph of order n and size m . Then
1 1 2
( 10 ( ))
2
( ) = ( 2( 1 )) ( 4) { [2( 1)( 1 ) 4( 1)]
8( 1)( 1)( 1 ) ( 1) }.
n m
DS T G
P n r n n r m
n m n r n r mn
Proof. The generalized xyz -point-line transformation graph T 10 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree n 1 r and the remaining
m vertices are with degree 2 by Theorem 1.1 and Theorem 1.2. Hence
10 2( 1 )( ) ( 1)
( ( )) =
( 1) 4( )
n n n m
m n m m
n r J I n r J
DS T G
n r J J I
Therefore,
10 ( 10 ( )) ( ) = | ( ( )) |
( 2( 1 )) 2( 1 ) ( 1)
= ( 1) ( 4) 4
DS T G
n n n m
m n m m
P I DS T G
n r I n r J n r J
n r J I J
Expanding the determinant, we get the required result. □
Theorem 3.4. Let G be an r regular graph of order n and size m . Then
1 1 2
( 11 ( ))
2
( ) = ( 2( 1 )) ( 2( 1)) { [2( 1)( 1 ) 2( 1)( 1)]
4( 1)( 1)( 1 )( 1) ( ) }.
n m
DS T G
P n r m n n r m m
n m n r m n m r mn
Proof. The generalized xyz -point-line transformation graph T 11 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree n 1 r and the remaining
m vertices are with degree m 1 by Theorem 1.1 and Theorem 1.2. Hence
11 2( 1 )( ) ( )
( ( )) =
( ) 2( 1)( )
n n n m
m n m m
n r J I n r m J
DS T G
n r m J m J I
Therefore,
11 ( 11 ( )) ( ) = | ( ( )) |
( 2( 1 )) 2( 1 ) ( )
= ( ) ( 2( 1)) 2( 1)
DS T G
n n n m
m n m m
P I DS T G
n r I n r J n m r J
n m r J m I m J
Expanding the determinant, we get the required result. □
Theorem 3.5. Let G be an r-regular graph of order n and size m . Then
1 1 2
( 1 ( ))
2
( ) = ( 2( 1 )) ( 4 ) { [( 1)2( 1 ) 4 ( 1)]
8( 1)( 1)( 1 ) ( 3 1) }.
n m
DS T G
P n r r n n r r m
n m n r r n r mn
Proof. The generalized xyz -point-line transformation graph T 1 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree n 1 r and the remaining
m vertices are with degree 2r by Theorem 1.1 and Theorem 1.2. Hence
1 2( 1 )( ) ( 3 1)
( ( )) =
( 3 1) 4 ( )
n n n m
m n m m
n r J I n r J
DS T G
n r J r J I
Therefore,
1
( 1 ( )) ( ) = | ( ( )) |
( 2( 1 )) 2( 1 ) ( 3 1)
= ( 3 1) ( 4 ) 4
DS T G
n n n m
m n m m
P I DS T G
n r I n r J n r J
n r J r I rJ
Expanding the determinant, we get the required result. □
Theorem 3.6. Let G be an r-regular graph of order n and size m . Then
1 1 2
( 1 ( ))
2
( ) = ( 2( 1 )) ( 2( 3 2 )) { [2( 1)( 1 )
2( 1)( 3 2 )] 4( 1)( 1)( 1 )( 3 2 )
( 2) }.
n m
DS T G
P n r m r n n r
m m r n m n r m r
n m r mn
Proof. The generalized xyz -point-line transformation graph T 1 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree R 1 = n 1 r and the remaining m vertices are with degree R 2 = m 3 2 r . Hence
1 1 2
1
1 2 2
2 ( ) ( )
( ( )) =
( ) 2 ( )
n n n m
m n m m
R J I R R J
DS T G
R R J R J I
Therefore,
1 ( 1 ( ))
1 1 1 2
1 2 2 2
( ) = | ( ( )) |
( 2 ) 2 ( )
= ( ) ( 2 ) 2
DS T G
n n n m
m n m m
P I DS T G
R I R J R R J
R R J R I R J
Expanding the determinant, we get the required result. □
Theorem 3.7. Let G be an r-regular graph of order n and size m . Then
1 1 2 2
( 1 ( ))
2
( ) = ( 4 ) ( 2( 1)) { [4( 1) 2( 1)]
8( 1)( 1)( 1) ( 2 1) }.
n m
DS T G
P r m n r m
n m m r m r mn
Proof. The generalized xyz -point-line transformation graph T 1 ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree 2r and the remaining m vertices are with degree m 1 . Hence
1 4 ( ) ( 2 1)
( ( )) =
( 2 1) 2( 1)( )
n n n m
m n m m
r J I m r J
DS T G
m r J m J I
Therefore,
1 ( 1 ( )) ( ) = | ( ( )) |
( 4 ) 4 ( 2 1)
= ( 2 1) ( 2( 1)) 2( 1)
DS T G
n n n m
m n m m
P I DS T G
r I rJ m r J
m r J m I m J
Expanding the determinant, we get the required result. □
Theorem 3.8. Let G be an r-regular graph of order n and size m . Then
1 1 2
( ( ))
2
( ) = ( 4 ) ( 2( 3 2 )) { [4( 1)
2( 1)( 3 2 )] 8( 1)( 1)( 3 2 ) ( 3) }.
n m
DS T G
P r m r n r
m m r n m m r r m mn
Proof. The generalized xyz -point-line transformation graph T ( ) G of a regular graph G of degree r has two types of vertices. The n vertices with degree 2r and the remaining m vertices are with degree m 3 2 r . Hence
4 ( ) ( 3)
( ( )) =
( 3) 2( 3 2 )( )
n n n m
m n m m
r J I m J
DS T G
m J m r J I
Therefore,
( ( )) ( ) = | ( ( )) |
( 4 ) 4 ( 3)
= ( 3) ( 2( 3 2 )) 2( 3 2 )
DS T G
n n n m
m n m m
P I DS T G
r I rJ m J
m J m r I m r J
Expanding the determinant, we get the required result. □
