A Study on Regularity in Vague Graphs

Vol. 5, No. 2, 2017, 71-81 ISSN: 2321 – 9238 (online) Published on 31 August 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/pindac.v5n2a2

71

Progress in

A Study on Regularity in Vague Graphs

Hossein Rashmanlou1, S.Firouzian2 and Mostafa Nouri Jouybari3

Department of Mathematics, Payame Noor University (PNU) P.O.Box 19395-3697, Tehran, Iran

Email: [email protected],[email protected] Corresponding author. Email: [email protected]

Received 1 May 2017; accepted 20 August 2017

Abstract. Theoretical concepts of graphs are highly utilized by computer scientists.Especially in research areas of computer science such as data mining, image segmentation, clustering image capturing and networking. The vague graphs are more flexible and compatible than fuzzy graphs due to the fact that they allowed the degree of membership of a vertex to an edge to be represented by interval valued in [0,1] rather than the crisp real values between 0 and 1. In this paper, some interesting properties of an edge regular vague graph are given. Also, new concepts such as strongly regular, edge regular, and biregular vague graphs are defined.

Keyword: Vague set, vague graph, biregular vague graph. AMS Mathematics Subject Classification (2010): 05C76

1.Introduction

72

graph. In this paper, some properties of an edge regular vague graph are given. Particularly, strongly regular, edge regular and biregular vague graphs are defined and the necessary and sufficient condition for a vague graph to be strongly regular is studied. Also, we have introduced a partially edge regular vague graph and fully edge regular vague graph with suitable illustrations.

2. Preliminaries

A graph is an ordered pair G =(V,E), where V is the set of vertices of G and E is the set of edges of G. A subgraph of a graph G=(V,E) is a graph H =(W,F), where W ⊆V and F⊆ E . A fuzzy graph G=(

σ

,

µ

) is a pair of functions

[0,1] :V →

σ

and

µ

:V×V →[0,1] with

µ

(u,v)≤

σ

(u)∧

σ

(v), for all u,v∈V , where V is a finite non-empty set and ∧ denote minimum.

A vague set A in an ordinary finite non-empty set X , is a pair (t_A, f_A), where [0,1]

:X →

t_A , f_A:X →[0,1] are true and false membership functions, respectively such that 0≤tA(x)+ fA(x)≤1, for all x∈X . Note that tA(x) is considered as the

lower bound for degree of membership of x in A and fA(x) is the lower bound for negative of membership of

x

in A. So, the degree of membership of

x

in the vague set

A, is characterized by the interval [t_A(x),1− f_A(x)].

Hence, a vague set is a special case of interval-valued sets studied by many mathematicians and applied in many branches of mathematics.

Let X and Y be two ordinary finite non-empty sets. We call a vague relation to be a vague subset of X×Y, that is an expression R defined by

} , | ) , ( ), , ( ), , ( {

= x y t x y f x y x X y Y

R _{R R} ∈ ∈

where t_R:X×Y→[0,1] , f_R:X×Y→[0,1] , which satisfies the condition 1

) , ( ) , (

0≤t_R x y + f_R x y ≤ , for all (x,y)∈X×Y. (See [8]).

Definition 2.1. [8] A vague graph is defined to be a pair G=(A,B), where )

, ( = t_A f_A

A is a vague set on V and B=(t_B, f_B) is a vague set on E⊆V×V such that t_B(xy)≤min(t_A(x),t_A(y)) and f_B(xy)≥max(f_A(x),f_A(y)), for each edge

E xy∈ .

The underlying crisp graph of a vague graph G =(A,B) , is the graph

) , ( = V E

G , where V ={v:tA(v)>0 and fA(v)>0} and

0} > }) , ({ 0, > }) , ({ : } , {{

= u v t u v f u v

E B B . V is called the vertex set and E is called

the edge set. A vague graph maybe also denoted as G=(V,E).

A vague graph G is said to be strong if tB(vivj)=min{tA(vi),tA(vj)} and )}

( ), ( { max = )

( i j A i A j

B vv f v f v

f , for every edge vivj∈E. A vague graph G is said to be complete if tB(vivj)=min{tA(vi),tA(vj)} and fB(vivj)=max{fA(vi), fA(vj)},

73 )

, ( = A B

G , where A=A=(tA, fA) and B=(tB, fB) is defined by:

)). ( ), ( ( max ) ( = ), ( )) ( ), ( ( min = )

(xy t x t y t xy f f xy f x f y

t_{B A A} − _{B B B} − _{A A}

Definition 2.2. [4] Let G =(V,E) be a vague graph. )

(i The neighborhood degree of a vertex v is defined as d (v)=(d (v),d (v))

f N t N

N ,

where

). ( =

) ( ) ( =

) (

) ( )

(

w f v

d and w t v

d A

v N w f

N A

v N w t

N

∑

∑

∈ ∈

)

(ii The degree of a vertex v_i is defined by d_G(v_i)=(d_t(v_i),d_f(v_i))=(k₁,k₂), where )

( =

) ( =

1 B i j j v i v i

t v t vv

d

k

∑

_≠ and ₂ = ( )= _B( _{i j})

j v i v i

f v f vv

d

k

∑

_≠ .

Definition 2.3. [4] A vague graph G =(V,E) is said to be −

) , ( )

(i k1 k2 regular if dG(vi)=(k1,k2), for all vi∈V and also G is said to be a

regular vague graph of degree (k₁,k₂). )

(ii bipartite if the vertex set V can be partitioned into two non-empty sets V₁ and V2 such that

0 = ) ( )

(a t_B v_iv_j and f_B(v_iv_j)=0, if (v_i,v_j)∈V₁ or (v_i,v_j)∈V₂ 0

= ) ( )

(b t_B v_iv_j , f_B(v_iv_j)>0, if vi∈V1 or vj∈V2

0 > ) ( )

(c t_B v_iv_j , f_B(v_iv_j)=0, if vi∈V1 or vj∈V2, for some i and j.

Definition 2.4. [16] Let G∗ =(V,E) be a crisp graph and let e=v_iv_j be an edge in ∗

G . Then, the degree of an edge e=v_iv_j∈E is defined as 2.

) ( ) ( = )

( _∗ + _∗ −

∗ i j _G i _G j G vv d v d v

d

3. Some properties of regularity in vague graphs Definition 3.1. Let G=(V,E) be a vague graph.

)

(i The degree of an edge e_ij∈E is defined as )

( 2 ) ( ) ( = )

( _{ij t i t j B i j}

t e d v d v t vv

d + −

or ( )= ( ) B( k j)

i k E j v k v k i B j k E k v i v ij

t e t vv t v v

d

∑

∑

≠ ∈ ≠

∈ +

) ( 2 ) ( ) ( = )

( _{ij f i f j B i j}

f e d v d v f vv

d + −

or ( )= ( ) B( k j)

i k E j v k v k i B j k E k v i v ij

f e f vv f vv

d

∑

∑

≠ ∈ ≠

∈ +

)

74

where

δ

t(G)=∧{dt(eij)|eij∈E} and

δ

f(G)=∧{df(eij)|eij∈E}.

)

(iii The maximum edge degree of G is ∆_E(G)=(∆_t(G),∆_f(G)), where ∆_t(G)=∨{d_t(e_ij)|e_ij∈E} and ∆_f(G)=∨{d_f(e_ij)|e_ij∈E}.

)

(iv The total edge degree of an edge e_ij∈E is defined as

) ( ) ( )

( =

)

( _{B k j B ij}

i k E j v k v k i B j k E k v i v ij

t e t vv t v v t e

td

∑

+

∑

+

≠ ∈ ≠

∈ ,

) ( ) ( )

( =

)

( _{B k j B ij}

i k E j v k v k i B j k E k v i v ij

f e f vv f v v f e

td

∑

+

∑

+

≠ ∈ ≠

∈ .

)

(v The edge degree of G is defined by d_G(e_ij)=(d_t(e_ij),d_f(e_ij)) and the total edge degree of G is defined by td_G(e_ij)=(td_t(e_ij),td_f(e_ij)).

Examples 3.2. Consider a vague graph G=(V,E) such that V ={u₁,u₂,u₃,u₄} and }

, , , , {

= u₁u₂ u₁u₄ u₂u₃ u₂u₄ u₃u₄

E .

Here, dt(e12)=0.5, df(e12)=1.9, dG(e12)=(0.5,1.9), tdt(e12)=0.5+0.2=0.7,

2.4 = 0.5 1.9 = ) (e₁₂ +

td_f . Hence, tdG(e12)=(0.7,2.4).

Definition 3.3. Let G=(V,E) be a vague graph. )

(i If each edge in G has the same degree (l1,l2), then G is said to be an edge regular vague graph.

)

(ii If each edge in G has the same total degree (t1,t2), then G is said to be a totally edge regular vague graph.

Theorem 3.4. Let G=(V,E) be a vague graph on a cycle G∗ =(V,E). Then )

( =

)

( _{G i j}

E j v i v i G V i

v d v

∑

d vv

∑

_{∈ ∈} .

Proof. Let G=(V,E) be a vague graph and G∗ be a cycle v₁v₂v_3⋯v_nv₁. Then ))

( ),

( (

= )

( 1 ₌₁ 1 ₌₁ 1

1

= +

∑

+

∑

+

∑

f i i

n i i i t n i i

i G n

i d vv d vv d vv . Now we have

1 1 1

3 2 2

1 1

1 =

= ),

( )

( ) ( = )

(vv d vv d vv d vv where v v

dt i i t t t n n

n

i

+

+ + + +

∑

⋯

) ( ) ( ) ( 2 ) ( ) (

=dt v1 +dt v2 − tB v1v2 +dt v2 +dt v3

) ( 2 ) ( ) ( )

(

2tB v2v3 + +dt vn +dt v1 − tB vnv1

− ⋯

) ( 2 )

( 2 ) ( 2

= d_t v₁ + d_t v₂ +_⋯+ d_t v_n −2(t_B(v₁v₂)+t_B(v₂v₃)+_⋯+t_B(v_nv₁))

) ( 2 ) ( 2

= 1

1 =

+

∈

∑

∑

− B i i

n

i i t V i v

v v t v

d = ( ) 2 ( ) 2 ( 1)

1 = 1 1

=

+ +

∈

∑

∑

∑

+ − B i i

n

i i i B n

i i t V i v

v v t v

v t v

75 ).

(

= t i

V i v v d

∑

∈

Similarly, ( ₁)= ( )

1

= f i i v_i V f i n

i d vv

∑

d v

∑

+ _∈ .

Hence, ₌₁ ( 1)=( ( ), ( ))= _V G( i)

i v i f V i v i t V i v i i G n

i d vv

∑

d v

∑

d v

∑

d v

∑

+ _{∈ ∈ ∈} .

Remark 3.5. Let G=(V,E) be a vague graph on a crisp graph G∗. Then, )), ( ) ( ), ( ) ( ( = )

( _{i j B i j}

G E j v i v j i B j i G E j v i v j i G E j v i v v v f v v d v v t v v d v v d _∗ ∈ ∗ ∈ ∈

∑

∑

∑

where d_G∗(vivj)=d_G∗(vi)+d_G∗(vj)−2, for all vivj∈E.

Theorem 3.6. Let G =(V,E) be a vague graph on a k−regular crisp graph G∗. Then, )). ( 1) ( ), ( 1) (( = )

( f i

V i v i t V i v j i G E j v i v v d k v d k v v

d

∑

∑

∑

∈ ∈ ∈ − −

Proof: By Remark 3.5 we have

)) ( ) ( ), ( ) ( ( = )

( _{i j B i j}

G E j v i v j i B j i G E j v i v j i G E j v i v v v f v v d v v t v v d v v d _∗ ∈ ∗ ∈ ∈

∑

∑

∑

)). ( 2) ) ( ) ( ( ), ( 2) ) ( ) ( ( (

= _G i _G j B i j

E j v i v j i B j G i G E j v i v v v f v d v d v v t v d v

d + − _∗ + _∗ −

∈ ∗

∗

∈

∑

∑

Since G∗ is a regular crisp graph, d v_i k

G∗( )= , for all vi∈V and we have

)), ( 2) ( ), ( 2) (( = )

( _{B i j}

E j v i v j i B E j v i v j i G E j v i v v v f k k v v t k k v v

d

∑

∑

∑

∈ ∈ ∈ − + − + )), ( 1) ),2( ( 1) (2( = )

( B i j

E j v i v j i B E j v i v j i G E j v i v v v f k v v t k v v

d

∑

∑

∑

∈ ∈ ∈ − − )). ( 1) ( ), ( 1) (( = )

( f i

V i v i t V i v j i G E j v i v v d k v d k v v

d

∑

∑

∑

∈ ∈ ∈ − −

Theorem 3.7. Let G=(V,E) be a vague graph on a crisp graph G∗. Then,

)). ( ) ( ) ( ), ( ) ( ) ( ( = )

( _{B i j}

E j v i v j i B j i G E j v i v j i B E j v i v j i B j i G E j v i v j i G E i v i v v v f v v f v v d v v t v v t v v d v v

td

∑

∑

∑

∑

∑

∈ ∗ ∈ ∈ ∗ ∈ ∈ + +

Proof: By definition of total edge degree of G, we have )) ( ), ( ( = )

( f i j

E j v i v j i t E j v i v j i G E j v i v v v td v v td v v

td

∑

∑

∑

∈ ∈ ∈ ))) ( ) ( ( )), ( ) ( ( (

= _{f i j B i j}

E j v i v j i B j i t E j v i v v v f v v d v v t v v

d +

∑

+

∑

76

)). ( )

( ),

( )

( (

= B i j

E j v i v j i f E j v i v j i B E j v i v j i t E j v i v

v v f v

v d v

v t v

v

d

∑

∑

∑

∑

∈ ∈

∈ ∈

+ +

By Remark 3.5, we get

), ( )

( ) ( (

= )

( _{B i j}

E j v i v j i B j i G E j v i v j i G E j v i v

v v t v

v t v v d v

v

td

∑

∑

∑

∈ ∗

∈ ∈

+

)). ( )

( )

( _{B i j}

E j v i v j i B j i G E j v i v

v v f v

v f v v

d

∑

∑

∈ ∗

∈

+

Theorem 3.8. Let G=(V,E) be a vague graph. Then (t_B, f_B) is a constant function if and only if the following are equivalent.

G i )

( is a edge regular vague graph. G

ii )

( is totally edge regular vague graph.

Proof: Assume that (t_B,f_B) is a constant function. Then t_B(v_iv_j)=c₁ and

2

= ) (vv c

f_{B i j} , for every v_iv_j∈E, where c₁ and c₂ are constants. Let G be a (l₁,l₂) -edge regular vague graph. Then, d_G(v_iv_j)=(l₁,l₂), for all v_iv_j∈E.

), ,

( = )) ( ) ( ), ( ) ( ( = )

(vv d vv t vv d vv f vv l₁ c₁ l₂ c₂

td_{G i j t i j} + _{B i j f i j} + _{B i j} + +

for all v_iv_j∈E which implies G is totally edge regular.

Let G be (t₁,t₂)−totally edge regular vague graph. Then td_G(v_iv_j)=(t₁,t₂), for all v_iv_j∈E. So, we have

). , ( = )) ( ) ( ), ( ) ( ( = )

(vv d vv t vv d vv f vv t₁ t₂ td_{G i j t i j} + _{B i j f i j} + _{B i j}

Now,

)) ( ),

( ( = )) ( ), (

(d_t v_iv_j d_f v_iv_j t₁−t_B v_iv_j t₂− f_B v_iv_j ).

, (

= t₁−c₁ t₂−c₂

Hence, G is (t₁−c₁,t₂−c₂) edge regular vague graph.

Conversely, assume that (i) and (ii) are equivalent. We have to prove that (tB, fB) is a constant function. Suppose that (tB, fB) is not a constant function. Then

) ( )

( _{i j B r s}

B vv t vv

t ≠ and f_B(v_iv_j)≠ f_B(v_rv_s) for at least one pair of v_iv_j,v_rv_s∈E. Let G be an (l₁,l₂) edge regular vague graph. Then, d_G(v_iv_j)=d_G(v_rv_s)=(l₁,l₂).

)) ( ) ( ), ( ) ( ( = )

( _{i j t i j B i j f i j B i j}

G vv d vv t vv d vv f vv

td + +

)), ( ),

( (

= l₁+t_B v_iv_j l₂+ f_B v_iv_j for all v_iv_j∈E and

)) ( ) ( ), ( ) ( ( = )

( _{r s t r s B r s f r s B r s}

G vv d vv t vv d vv f vv

td + +

)), ( ),

( (

= l1+tB vrvs l2+ fB vrvs

77

Since, tB(vivj)≠tB(vrvs) and fB(vivj)≠ fB(vrvs) we have tdG(vivj)≠tdG(vrvs).

Hence, G is not a totally edge regular that is contradiction to our assumption. Therefore, )

,

(t_B f_B is a constant function. Similarly we can show that (t_B, f_B) is a constant function when G is a totally edge regular vague graph.

Theorem 3.9. Let G =(V,E) be a vague graph on a k−regular crisp graph G∗. Then, )

,

(tB fB is a constant if and only if G is both regular and edge regular vague graph.

Proof. Let G=(V,E) be a vague graph on G∗ and let G∗ be a k−regular crisp graph. Assume that t_B and f_B are constant functions, i.e., t_B(v_iv_j)=c and

t v v

f_B( _{i j})= , for all v_iv_j∈E, where c,t are constants. By definition of degree of a vertex we have

)) ( ), ( ( = )

( _{i t i f i}

G v d v d v

d =( ( ), _B( _{i j}))

E j v i v j i B E j v i v

v v f v

v

t

∑

∑

∈ ∈

), , (

= c t

E j v i v E j v i v

∑

∑

∈ ∈

for all vi∈V. Hence, dG(vi)=(kc,kt). Therefore, G is regular vague graph. Now,

)) ( ), ( ( = )

( _{i j t i j f i j}

G vv td vv td vv

td , where

) ( ) ( )

( = )

( i j B i k B k j B i j

t vv t vv t v v t vv

td

∑

+

∑

+ c c c

i k E j v k v j k E k v i v

+

+

∑

∑

≠ ∈ ≠

∈ =

c k

c k

c( −1)+ ( −1)+

= =c(2k−1).

Similarly, td_f(v_iv_j)=t(2k−1) , for all v_iv_j∈E . Hence, G is also totally edge regular vague graph.

Conversely, assume that G is both regular and edge regular vague graph. We prove that )

,

(tB fB is a constant function. Since G is regular, dG(vi)=(c1,c2), for all vi∈V .

Also, G is totally edge regular so, td_G(v_iv_j)=(t₁,t₂), for all v_iv_j∈E.

By definition of totally edge degree we have td_G(v_iv_j)=(td_t(v_iv_j),td_f(v_iv_j)), where )

( ) ( ) ( = )

( _{i j t i t j B i j}

G vv d v d v t vv

td + − , for all v_iv_j∈E , t₁=c₁+c₂−t_B(v_iv_j) . So,

1 1

2 = )

(vv c t

t_{B i j} − . Similarly we have fB(vivj)=2c2−t2, for all vivj∈E . Hence,

) ,

(tB fB is a constant function.

Definition 3.10. A vague graph G=(V,E), where V ={v₁,v₂,_⋯,v_n} is said to be strongly regular, if it satisfies the following axioms:

G i )

( is k=(k₁,k₂)−regular vague graph )

(ii The sum of membership values and non-membership values of the common

78

Note 1. Any strongly vague graph G is denoted by G=(n,k,

λ

,

δ

).

Examples 3.11. Consider a vague graph G =(V,E) where V ={u₁,u₂,u₃,u₄} and }

, , , , , {

= u1u2 u2u4 u3u4 u1u3 u2u3 u1u4

E .

Here, n=4, k=(k₁,k₂)=(0.3,1.5),

λ

=(

λ

₁,

λ

₂)=(0.3,0.8),

δ

=(

δ

₁,

δ

₂)=(0,0). So, G is strongly regular vague graph.

Theorem 3.12. If G=(V,E) is a complete vague graph with (t_A, f_A) and (t_B, f_B) as constant functions, then G is a strongly regular vague graph.

Proof: Let G =(V,E) be a complete vague graph where V ={v₁,v₂,_⋯,v_n}. Since

B A A f t

t , , and fB are constant functions. That is, tA(vi)=r , fA(vi)=s , for all

V

v_i∈ and t_B(v_iv_j)=c and f_B(v_iv_j)=t , for all v_iv_j∈E where r,s,c,t are constants. To prove that G is a strongly regular vague graph, we have to show that G is

− ) , ( = k₁ k₂

k regular vague graph and the adjacent vertices have the same common neighborhood

λ

=(

λ

₁,

λ

₂) and non-adjacent vertices have the same common neighborhood

δ

=(

δ

1,

δ

2). Now,

)) ( ), ( ( = )

( _{i t i f i}

G v d v d v

d =( ( ), B( i j))

E j v i v j i B E j v i v

v v f v

v

t

∑

∑

∈ ∈

). (

) 1) ( , 1) ((

= n− c n− t SinceGiscomplete

Hence, G is an ((n−1)c,(n−1)t)− regular vague graph. The sum of membership values and non-membership values of common neighborhood vertices of any pair of adjacent vertices

λ

=((n−2)r,(n−2)s) are the same and the sum of membership values and non-membership values of common neighborhood vertices of any pair of non-adjacent vertices

δ

=0, since G is complete vague graph. So we have the proof.

Remark 3.13. If G is a strongly regular disconnected vague graph then,

δ

=0.

Definition 3.14. A vague graph G=(V,E) is said to be a biregular vague graph if it satisfies the following axioms:

G i )

( is k=(k1,k2)−regular vague graph.

2 1

= )

(ii V V ∪V be the bipartition of V and every vertex in V₁ has the same neighborhood degree M =(M1,M1) and every vertex in V2 has the same neighborhood

degree N =(N1,N2), where M and N are constants.

Theorem 3.15. If G =(V,E) is a strongly regular vague graph which is strong then, G is (k1,k2)−regular.

79 −

) ,

(k₁ k₂ regular. Since G is strong, we have

  

∈/ ∈

E v v all for v

t v t

E v v all for v

v t

j i j

A i A

j i j

i B

)) ( ), ( ( min 0 = ) (

  

∈/ ∈

. ))

( ), ( ( max 0 = ) (

E v v all for v

f v f

E v v all for v

v f

j i j

A i A

j i j

i B

Now the degree of a vertex vi in G is dG(vi)=(dt(vi),df(vi)), where

, = ) ( ) ( = ) ( = )

(v t vv t v t v k₁

d _{A i A j}

j v i v j i B j v i v i

t

∑

∑

∧

≠ ≠

for all v_i∈V and

, = ) ( ) ( =

) ( =

)

(v f vv f v f v k2

d A i A j

j v i v j i B j v i v i

f

∑

∑

∧

≠ ≠

for all vi∈V . (Since G is strong)

Hence, dG(vi)=(k1,k2), for all vi∈V . So, G is a (k1,k2)-regular vague graph.

Theorem 3.16. Let G=(V,E) be a strong vague graph. Then G is a strongly regular if and only if G is a strongly regular.

Proof: Assume that G=(V,E) is a strongly regular vague graph. Then by definition, we

have G is (k1,k2)−regular and the adjacent vertices and the non-adjacent vertices have the same common neighborhood

λ

=(

λ

₁,

λ

₂) and

δ

=(

δ

₁,

δ

₂), respectively. We have to prove that G is a strongly regular vague graph. If G is strongly regular vague graph

which is strong then G is a (k₁,k₂)−regular vague graph by Theorem 3.17. Next, let S₁ and S₂ be the sets of all adjacent vertices and non-adjacent vertices of G. That is,

} |

{ =

1 vv vv E

S _{i j i j}∈ , where v_i and v_j have same common neighborhood )

, ( =

λ

1

λ

2

λ

and S₂ ={v_iv_j|v_iv_j∈/E} , where vi and vj have same common

neighborhood

δ

=(

δ

₁,

δ

₂). Then, S₁={v_iv_j|v_iv_j∈E}, where v_i and v_j have same common neighborhood

δ

=(

δ

1,

δ

2) and S2 ={vivj|vivj∈/E} , where vi and vj

have same common neighborhood

λ

=(

λ

1,

λ

2). Which implies G is a strongly regular. Similarly we can prove the converse.

Theorem 3.17. A strongly regular vague graph G is a biregular vague graph if the adjacent vertices have the same common neighborhood

λ

=(

λ

1,

λ

2)≠0 and the non-adjacent vertices have the same common neighborhood

δ

=(

δ

₁,

δ

₂)≠0.

80 )

, ( = )

(v k1 k2

d i , for all vi∈V . Assume that the adjacent vertices have the same common

neighborhood

δ

=(

δ

1,

δ

2)≠0. Let S be the sets of all non-adjacent vertices. That is |

{ = v_iv_j

S v_i is not adjacent to v_j,i≠ j,v_i,v_j∈V}. Now the vertex partition of G is }

| { =

1 v v S

V i i∈ and V2 ={vj|vj∈S}. Then V1 and V2 have the same neighborhood

degree, since G is a strongly regular. Hence, G is a biregular vague graph.

4. Conclusion

Graph theory is an extremely useful tool in solving the combinatorial problemsin different areas including geometry, algebra, number theory, topology, operations research, optimization, and computer science. The concept of vague sets is due toGau and Buehrer who studied the concept with the aim of interpreting the real lifeproblems in better way than the existing mechanisms such as Fuzzy sets.In this paper,novel properties of an edge regular vague graph are given. Likewise, new concepts such as strongly regular, edge regular, and biregular vague graphs are defined.

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