International Journal of Analysis and Applications
Volume 19, Number 1 (2021), 47-64
URL: https://doi.org/10.28924/2291-8639
DOI: 10.28924/2291-8639-19-2021-47
ON DOMINATION TOPOLOGICAL INDICES OF GRAPHS
A.M. HANAN AHMED1,∗, ANWAR ALWARDI2, M. RUBY SALESTINA1
1Department of Mathematics, Yuvaraja’s college, University of Mysore, Mysuru, India
2Department of Mathematics, University of Aden, Yemen
∗Corresponding author: hananahmed1a@gmail.com
Abstract. Topological indices and domination in graphs are the essential topics in the theory of graphs.
A set of vertices D ⊆ V (G) is said to be a dominating set for G if any vertex v ∈ V −D is adjacent to some
vertex u ∈ D. In this research work, we define a new degree of each vertex v ∈ V (G), called the domination
degree of v and denoted by dd(v), along with this new degree some domination indices based on domination
degree are introduced. We study some basic properties of the domination degree function. Exact values and
bounds for domination Zagreb indices of some families of graphs including the join and corona product are
obtained. Finally, we generalize the domination degree of the vertex and new general indices are defined.
1. Introduction
In this research article, we assume that G = (V,E) is a connected simple graph. In the field of chemistry,
graph theory has provided many useful tools, such as topological indices. Chem-informatics is one of the
latest concepts which is a join of chemistry, mathematics, and information science.
Topological indices are numerical parameters of the graph, such that these parameters are the same for the
Received September 21st, 2020; accepted October 20th, 2020; published November 24th, 2020.
2010 Mathematics Subject Classification. 05C69, 05C90, 05C35.
Key words and phrases. domination Zagreb indices; domination degree; minimal dominating set; total number of minimal
domination sets.
©2021 Authors retain the copyrights
of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
47
https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-47
Int. J. Anal. Appl. 19 (1) (2021) 48
graph which they are isomorphism. Some of the major classes of topological indices are distance based-
topological indices (see [2], [3]) and degree- based-topological indices (see [5], [20], [23]). Degree based-
topological indices are of great significance. The Wiener index W(G) is the old index and the first distance-
based, introduced by chemist Wiener [24] in 1974. After the introduction of the Wiener index, many
another distance-based topological indices, have been proposition and take into consideration in chemical
and mathematical chemical literature. For example, Harary index [17] and eccentric connectivity index [22].
presently a great number of vertex-degree-based graph invariants are being studied in mathematical and
mathematical chemical literature ( [8], [9]). Among them, the Zagreb indices M1(G) and M2(G) are the
most widely investigated. Those have been inserted more than forty years ago ( [11], [12]), which are defined
as follows:
M1(G) =
∑
v∈V (G)
d2(v) =
∑
uv∈E(G)
d(u) + d(v)
M2(G) =
∑
uv∈E(G)
d(u)d(v).
For properties of the two Zagreb indices see [10] and the papers cited therein. The Zagreb co-indices defined
in [21], and are given by :
M1(G) =
∑
d(u) + d(v) and M2(G) =
∑
d(u)d(v), where uv is not an edge in E(G). The degree of a
vertex u in G, d(u) is the number of edges that are incident to u in G. The maximum and minimum degrees of
vertices of a graph G are denoted by ∆(G) and δ(G) respectively. G is the complement of a graph G, having
the same vertex set of G so that two vertices of G are neighboring if and only if they are not neighboring
in G. If for every two vertices u,v ∈ V, there exists a (u,v)-path in G, then G is connected, otherwise, G is
said to be disconnected. A set D ⊆ V is said to be a dominating set of G, if for any vertex v ∈ V −D there
exists a vertex u ∈ D such that u and v are adjacent. The domination number γ(G) of G is the minimum
cardinality of a minimal dominating set in G. The upper domination number Γ(G) of G is the maximum
cardinality of a minimal dominating set in G [4]. For a survey of domination in graphs, refer to ( [14], [15]).
A dominating-set D = {v1,v2, ...,vr} is minimal if D−vi is not a dominating set. We use Tm to denote the
number of minimal dominating sets. In [7] a graph has at most O(1.7159n) minimal dominating-sets and
there exist graphs with at leastO(1.5705n) minimal dominating-sets. For more definitions or properties, we
refer to ( [7], [18], [19]).
2. Domination Degree in Graphs
In this partition, we sitting the definition of domination degree of the vertex v. We consider the lower
and upper bound for this degree, and we study some basic properties of the domination degree.
Definition 2.1.
Int. J. Anal. Appl. 19 (1) (2021) 49
For each vertex v ∈ V (G), the domination degree denoted by dd(v) and defined as the number of minimal
dominating sets of G which contains v.
The minimum domination and maximum domination degree of G are denoted by δd(G) = δd and ∆d(G) =
∆d respectively, where
δd = min{dd(v) : v ∈ V (G)} and ∆d = max{dd(v) : v ∈ V (G)}.
Let v ∈ V (G) and v
′
∈ G
′
. Then dd G(v) = dd G′ (v
′
) if G ∼= G
′
.
The domination degree function is obviously invariant under isomorphism.
Observation 2.1.
1 ≤ dd(v) ≤ Tm(G), where Tm(G) denotes the total number of minimal dominating sets.
Observation 2.2.
Suppose G is a graph of n ≥ 2 vertices having ∆(G) = n − 1. Then γ(G) = 1 and dd(v) ≥ 1, for any
v ∈ V (G). Also, dd(v) = 1 if and only if d(v) = n− 1.
Corollary 2.1.
If G ∼= Kn and G is the complement of G, then
dd G(v) = dd G(v), and Tm(G) = 1.
Observation 2.3.
Let G(V,E) be a graph with minimal dominating sets S1,S2, ...,St. Then
tγ(G) ≤
∑
v∈V (G) dd(v) ≤ tΓ(G).
We use the notion ρ(G) =
∑
v∈V (G) dd(v).
Proposition 2.1.
Let G be the complete bipartite graph Kr,s. Then dd G(v) = dG(v).
Proof. Let the bipartite sets of Kr,s be A and B, where A contains the vertices of degree s and B contains
the vertices of degree r. Also Tm(G) = rs, such that if v ∈ A ⇒ dd G(v) = s. similarly if v ∈ B ⇒
dd G(v) = r. �
Observation 2.4.
Let G =
⋃t
i=1 Gi be the disjoint union of graphs G1, G2,...,Gt. Then γ(G) =
∑t
i=1 γ(Gi) and Tm(G) =∏t
i=1 Tm(Gi). For v ∈ V (Gi), dd G(v) = dd Gi (v)
∏t
j=1 Tm(Gj), j 6= i.
Proposition 2.2.
Given H is a spanning subgraph of G with V (H) is the same as V (G). If the domination number of H is
the same as the domination number of G, then Tm(H) ≤ Tm(G).
Int. J. Anal. Appl. 19 (1) (2021) 50
Proof. Use the first presumption, every dominating set for H is also a dominating set of G. As γ(H) = γ(G),
its ensured that every minimal dominating set of minimum cardinality for H is also a minimal dominating
set of minimum cardinality for G. �
Definition 2.2.
The graph G is called k−domination regular graph if and only if dd(v) = k for all v ∈ V (G).
Example 2.1. Sr and Kn are 1−domination regular graph.
Proposition 2.3.
Let G be the double star graph Sr,s . Then
Tm(G) = 4, and any double star graph is 2−domination regular graph.
Proof. Let {v,v1, ...,vr−1,w,w1, ...,ws−1} be the set of all vertices of G with {v,w} be the center ver-
tices. There are four type of minimal dominating sets as following: {v,w}, {v1,v2, ...,vr−1,w1,w2, ...,ws−1},
{v,w1,w2, ...,ws−1} and {w,v1,v2, ...,vr−1}.
⇒ Tm(G) = 4. From this we get, dd(v) = 2 for all v ∈ V (G). �
3. Domination Zagreb Indices of a graph
Definition 3.1.
Let G be a simple connected graph, the first domination, second domination Zagreb and modified first Zagreb
indices are define as :
DM1(G) =
∑
v∈V (G)
d2d(v) ,
DM2(G) =
∑
uv∈E(G)
dd(u)dd(v) ,
DM∗1 =
∑
uv∈E(G)
[dd(u) + dd(v)] .
Lemma 3.1.
Tm(Sr) = 2 and Tm(Kn) = n.
For all v ∈ V (Sr) or v ∈ V (Kn), we get dd(v) = 1.
Proposition 3.1.
(1) For the star graph Sr, with r + 1 vertices.
DM1(Sr) = r + 1 , DM2(Sr) = r and DM
∗
1 (Sr) = 2r.
(2) For the complete graph Kn
DM1(Kn) = n , DM2(Kn) =
n(n− 1)
2
and DM∗1 (Kn) = n(n− 1).
Int. J. Anal. Appl. 19 (1) (2021) 51
(3) For the double star graph Sr,s
DM1(Sr,s) = 4(r + s + 2) DM2(Sr,s) = 4(r + s + 1) and DM
∗
1 (Sr,s) = 4(r + s + 1).
Lemma 3.2.
Tm(Kr,s) = rs + 2 and dd(v) =
r + 1;s + 1.
for all v ∈ V (Kr,s).
Theorem 3.1.
If G ∼= Kr,s, then
DM1(G) = r(r + 1)
2 + s(s + 1)2 ,
DM2(G) = (r + 1)(s + 1)rs,
DM∗1 (G) = rs(r + s + 2).
Proof. By using the definition of domination Zagreb indices and lemma 3.2, we get the results. �
Corollary 3.1.
Let G be the complete bipartite Kr,s. Then
(1) DM1(G) = M1(G) + 4rs + (r + s).
(2) DM2(G) = M2(G) + M1(G) + rs.
(3) DM∗1 (G) = M1(G) + 2rs.
Proof. In the complete bipartite Kr,s, we can see that dd(vi) = d(vi) + 1 for all i = 1, 2, ...,r + s
DM1(G) =
∑
v∈V (G)
d2d(v) =
∑
v∈V (G)
(d(v) + 1)2
= M1(G) + 4rs + (r + s)
And,
DM2(G) =
∑
uv∈E(G)
dd(u)dd(v) =
∑
uv∈E(G)
(d(u) + 1)(d(v) + 1)
= M2(G) + M1(G) + rs,
DM∗1 (G) =
∑
uv∈E(G)
[dd(u) + dd(v)] =
∑
uv∈E(G)
[(d(u) + 1) + (d(v) + 1)]
= M1(G) + 2rs.
�
Proposition 3.2.
Int. J. Anal. Appl. 19 (1) (2021) 52
If G ∼= Kr,s, then
(1) DM1(G) = DM
∗
1 (G) = M1(G).
(2) DM2(G) = M2(G).
Proof. See proposition 2.1. �
Lemma 3.3.
Let G be the Windmill graph Wdsr. Then Tm(Wd
s
r) = (r − 1)s + 1. And
dd(v) =
1, if v is the center vertex;(r − 1)s−1, otherwise.
Theorem 3.2.
If G ∼= Wdsr, then
DM1(G) = 1 + s(r − 1)2s−1,
DM2(G) = s((r − 1)s + (r − 1)2s−1(
r − 2
2
)),
DM∗1 (G) = s(r − 1)(1 + (r − 1)
s).
Proof.
DM1(G) =
∑
v∈V (G)
d2d(v) = 1 +
∑
v∈V (G)−1
d2d(v)
= 1 + (r − 1)2(s−1)(|V (G)|− 1)
= 1 + s(r − 1)2s−1.
Let E1 denote the set of all edges which are incident with the center vertex and E2 be the set of all edges of
the complete graph, then
DM2(G) =
∑
uv∈E(G)
dd(u)dd(v) =
∑
uv∈E1
dd(u)dd(v) +
∑
uv∈E2
dd(u)dd(v)
= (r − 1)s−1|E1| + s(r − 1)2s−2|E2|
= s((r − 1)s + (r − 1)2s−1(
r − 2
2
)),
DM∗1 (G) =
∑
uv∈E(G)
[dd(u) + dd(v)] =
∑
uv∈E1
[dd(d) + dd(v)] +
∑
uv∈E2
[dd(u) + dd(v)]
= (1 + (r − 1)s−1)|E1| + 2s(r − 1)s−1|E2|
= s(r − 1)(1 + (r − 1)s).
�
Int. J. Anal. Appl. 19 (1) (2021) 53
Proposition 3.3.
If G is K−domination regular graph with n vertices, and m edges, then
DM1(G) = nK
2, DM2(G) = mK
2 and DM∗1 (G) = 2mK.
Definition 3.2.
Let P3 be the 3 vertex tree, is rooted in one of its terminal vertices . For k = 2, 3, 4, ... build the rooted tree
Bk by identifying the roots of k-copies of P3. The vertex obtained by identifying the roots of P3−trees is the
root of Bk [16].
Definition 3.3.
Let d ≥ 2 be an integer. Let β1,β2, ...,βd be as specified in Definition 3.2 i.e., β1,β2, ...,βd ∈ {B2,B3, ...}.
A Kragujevac tree T is a tree has a vertex of degree d, neighboring to the roots of β1,β2, ...,βd. This vertex
be the central- vertex of T , where d is the degree of the tree T . The subgraphs β1,β2, ...,βd are the branches
of T . Recall that some (or all) branches of T may be mutually isomorphic [16].
The branch Bk has 2k + 1 vertices. Therefore, if in the Kragujevac tree T, specified in Definition 3.3,
βi ∼= βki , i = 1, 2, ...,d then its order is n(T) = 1 +
∑d
i=1(2ki + 1).
Proposition 3.4.
Let T be the Kragujevac tree of order n(T) = 1 +
∑d
i=1(2ki + 1) and size m. Then
DM1(T) = 4[1 +
d∑
i=1
(2ki + 1)],
DM2(T) = DM
∗
1 (T) = 4m.
Proof. It is easy to see that, in the Kragujevac tree of order n(T) = 1 +
∑d
i=1(2ki + 1) and size m there are
four types of minimal dominating sets. The set which contains the center vertex and all pendent vertices, the
set which contains the center vertex and all vertices adjacent to the pendent vertices, the set which contains
the roots of β1,β2, ...,βd and all pendent vertices, and the set which contains the roots of β1,β2, ...,βd and
all vertices adjacent to the pendent vertices. Hence, dd(v) = 2 for all v ∈ V (T). So by using the definition
of domination Zagreb indices we get the result. �
Definition 3.4.
Let G1 and G2 be any two graphs. The Cartesian product G1 ×G2 is defined as [6] the graph has vertex set
(V (G1) ×V (G2)) such that any two vertices u = (u1,u2) and v = (v1,v2) are adjacent if and only if either
([u1 = v1 and {u2,v2}∈ E(G2)]) or ([u2 = v2 and {u1,v1}∈ E(G1)]).
Definition 3.5.
Int. J. Anal. Appl. 19 (1) (2021) 54
Book graph Br is a Cartesian product of a star and single edge Sr+1 × P2. The generalization of the book
graph to n “stacked“ is the (r,s)−Stacked book graph [13].
Lemma 3.4.
If G ∼= Br, then Tm(G) = 2r + 3. Further, for any vertex v ∈ V (Br)
dd(v) =
3, if v is the center vertex;2r−1 + 1, otherwise.
Proof. Let uv be the center edge in book graph such that {u,v} is the set of center vertices. Let {v1,v2, ...,vr}
be the set of all vertices which are adjacent with the center vertex v. Similarly {u1,u2, ...,ur} be the set of
all vertices which are adjacent with the center vertex u. There are four types of minimal dominating sets.
First type is{u,v}. Second type is {v,u1,u2, ...,ur} and {u,v1,v2, ...,vr}. Third type is {u,u1,u2, ...,ur}
and {v,v1,v2, ...,vr}. Fourth type is only those minimal dominating sets which are formed by taking one
vertex from each section other than u and v. So there are 2r − 2 minimal dominating sets of fourth type.
Hence, Tm(Br) = 2
r + 3 and for all v ∈ V (Br) we get
dd(v) =
3, if v is the center vertex;2r−1 + 1, otherwise. �
Theorem 3.3.
Let G be a book graph Br where r ≥ 3. Then
DM1(Br) = 2r(2
r−1 + 1)2 + 18 ,
DM2(Br) = r(2
r−1 + 1)[2r−1 + 7] + 9 ,
DM∗1 (Br) = 2
r+1r + 2r(4 + 2r−1) + 6 .
Proof.
DM1(Br) =
∑
w∈V (Br)
d2d(w) =
∑
w∈V (Br−{u,v})
(2r−1 + 1)2 +
∑
w∈{u,v}
32
= 2r(2r−1 + 1)2 + 18.
There are three type of edges in the book graph.
Let E1 denote the set of r edges (uivi) with initial and terminal vertices of the same domination degree
2r−1 + 1, E2 denote the set containing only one edge (uv) with initial and terminal vertices of the same
domination degree 3, and E3 denote the set of 2r edges of initial vertices of the domination degree 3 and
terminal vertices of domination degree 2r−1 + 1. Hence,
Int. J. Anal. Appl. 19 (1) (2021) 55
DM2(Br) =
∑
uv∈E(Br)
dd(u)dd(v)
=
∑
uv∈E1
(2r−1 + 1)2 +
∑
un∈E2
9 +
∑
uv∈E3
3(2r−1 + 1)
= r(2r−1 + 1)[2r−1 + 7] + 9 ,
DM∗1 (Br) =
∑
uv∈E(Br)
[dd(u) + dd(v)]
=
∑
uv∈E1
[(2r−1 + 1) + (2r−1 + 1)] +
∑
uv∈E2
[3 + 3] +
∑
uv∈E3
[3 + 2r−1 + 1]
= 2r+1 + 2r(4 + 2r−1) + 6.
�
Lemma 3.5.
Let G ∼= Kn1,n2,...,nk . Then
Tm(G) =
k∑
i=2
n1ni +
k∑
i=3
n2ni + ... + nk−1nk + k .
Theorem 3.4.
If G ∼= Kn1,n2,...,nk , then
DM1(G) = M1(G) + 4(Tm(G) −k) +
k∑
i=1
ni ,
DM2(G) = M2(G) + M1(G) + Tm(G) −k ,
DM∗1 (G) = M1(G) + 2(Tm(G) −k) .
Proof. Suppose G ∼= Kn1,n2,...,nk , note that for any vertex v ∈ G we have dd(v) = d(v) + 1, and |E(G)| =
Tm(G) −k. So, by the definition of domination Zagreb indices we get the result. �
Lemma 3.6.
For any connected graph G with n1 vertices and m1 edges. Let H ∼= G◦Kn2 , where Kn2 is the complete graph
of n2 vertices and m2 edges. There are (n2 + 1)
n1 minimal domination sets in H, and dd(v) = (n2 + 1)
n1−1.
Theorem 3.5.
For any connected graph G of n1 vertices and m1 edges, we have
DM1(G◦Kn2 ) = (n1 + n1n2)(n2 + 1)
2(n1−1) ,
DM2(G◦Kn2 ) = (n2 + 1)
2(n1−1)[2m1 + n2(n2 + 2n1 − 1)] ,
DM∗1 (G◦Kn2 ) = 4(n2 + 1)
n1−1[2m1 + n2(n2 + 2n1 − 1)] .
Int. J. Anal. Appl. 19 (1) (2021) 56
Proof. Note that |V (G◦Kn2 )| = n1 + n1n2. Hence, by the definition of first domination Zagreb indices and
Lemma 3.6, we get
DM1(G◦Kn2 ) = (n1 + n1n2)(n2 + 1)
2(n1−1).
There are three types of edges in G◦Kn2 . All edges of G, all edges of Kn2 and let E1 denote the set of all
edges that connect vertex from G and vertex from Kn2 . So, we have
DM2(G◦Kn2 ) =
∑
uv∈E(GoKn2 )
dd(u)dd(v)
=
∑
uv∈E(G)
dd(u)dd(v) +
∑
uv∈E(Kn2 )
dd(u)dd(v) +
∑
uv∈E1
dd(u)dd(v)
= m1(n2 + 1)
2(n1−1) + (n2 + 1)
2(n1−1)|E(Kn2 )| + n1n2(n2 + 1)
2(n1−1)
= (n2 + 1)
2n1−2[m1 +
n2(n2 − 1)
2
+ n1n2]
= (n2 + 1)
2(n1−1)[2m1 + n2(n2 + 2n1 − 1)],
DM∗1 (G◦Kn2 ) =
∑
uv∈E(G◦Kn2 )
[dd(u) + dd(v)]
=
∑
uv∈E(G)
2(n2 + 1)
n1−1 +
∑
uv∈E(Kn2 )
2(n2 + 1)
n1−1 +
∑
uv∈E1
2(n2 + 1)
n1−1
= 4(n2 + 1)
n1−1[2m1 + n2(n2 + 2n1 − 1)].
�
Lemma 3.7.
Let H ∼= G◦Kn2 , where G be any connected graph of order n1. Then
Tm(H) =
n1∑
i=0
(
n1
i
)
.
Theorem 3.6.
If G be a graph of order n1 and size m1. Let H ∼= G◦Kn2 then
DM1(H) = (Tm(H) − 2n1−1)2(n1 + n1n2) ,
DM2(H) = (Tm(H) − 2n1−1)2(m1 + n1n2) ,
DM∗1 (H) = (2Tm(H) − 2n1 )(m1 + n1n2) .
Proof. For any vertex v ∈ V (H), it is not easy to see that H ∼= G◦Kn2 is domination regular graph. Every
v ∈ V (H) is contained in every minimal dominating sets of H except
Int. J. Anal. Appl. 19 (1) (2021) 57
(
n1−1
0
)
+
(
n1−1
1
)
+ ... +
(
n1−1
n1−2
)
+
(
n1−1
n1−1
)
= 2n1−1 minimal dominating sets.
Hence, dd H (v) = Tm(H) − 2n1−1 and
DM1(H) = (Tm(H) − 2n1−1)2(n1 + n1n2) ,
DM2(H) = (Tm(H) − 2n1−1)2(m1 + n1n2) ,
DM∗1 (H) = (2Tm(H) − 2n1 )(m1 + n1n2) .
�
A join of two graphs G1 and G2 is denoted by G1 + G2, with disjoint vertex sets V1 and V2 is the graph
on the vertex set V1 ∪V2 and the edge set E1 ∪E2 ∪{u1u2 : u1 ∈ V1,u2 ∈ V2} [1].
Lemma 3.8.
Let G1 and G2 be any non complete graphs of n1, n2 vertices respectively. Then
Tm(G1 + G2) = Tm(G1) + Tm(G2) + n1n2, and
dd G1+G2 (v) =
dd G1 (v) + n2, if v ∈ V (G1);dd G2 (v) + n1, if v ∈ V (G2).
Proof. There are three types of minimal dominating sets in G1 + G2 graph:
The minimal-dominating sets of G1, all the minimal dominating sets of G2 and the sets of size two of all
minimal dominating sets containing one vertex from G1 and another vertex from G2.
Hence, Tm(G1 + G2) = Tm(G1) + Tm(G2) + n1n2, and
dd G1+G2 (v) =
dd G1 (v) + n2, if v ∈ V (G1);dd G2 (v) + n1, if v ∈ V (G2). �
Theorem 3.7.
Let G1 and G2 be any non complete graphs having n1, n2 vertices and m1, m2 edges respectively. Then
(1)
DM1(G1 + G2) = DM1(G1) + DM2(G2) + 2n2ρ(G1)
+ 2n1ρ(G2) + n1(n
2
2 + n2n1),
(2)
DM2(G1 + G2) = DM2(G1)(1 + n2) + m1n
2
2 + DM2(G2)(1 + n1) + m2n
2
1
+ [n1n2 + ρ(G1)][n1n2 + ρ(G2)],
Int. J. Anal. Appl. 19 (1) (2021) 58
(3)
DM∗1 (G1 + G2) = DM
∗
1 (G1) + DM
∗
1 (G2) + 2m1n2 + 2n1m2
+ (ρ(G2) + n2(n1 + 1))(ρ(G1) + n1n2).
Proof.
DM1(G1 + G2) =
∑
v∈V (G1+G2)
d2d G1+G2 (v)
=
∑
v∈V (G1)
(dd G1 (v) + n2)
2 +
∑
v∈V (G2)
(dd G2 (v) + n1)
2
=
∑
v∈V (G1)
dd G1 (v)
2 + 2n2
∑
v∈V (G1)
dd G1 (v) + n
2
2
∑
v∈V (G1)
1
+
∑
v∈V (G2)
dd G2 (v)
2 + 2n1
∑
v∈V (G2)
dd G2 (v) + n
2
1
∑
v∈V (G1)
1
= DM1(G1) + DM2(G2) + 2n2
∑
v∈V (G1)
dd G1 (v)
+ 2n1
∑
v∈V (G2)
dd G2 (v) + n1(n
2
2 + n2n1)
= DM1(G1) + DM2(G2) + 2n2ρ(G1)
+ 2n1ρ(G2) + n1(n
2
2 + n2n1).
And,
DM2(G1 + G2) =
∑
uv∈E(G1+G2)
dd G1+G2 (u)dd G1+G2 (v)
=
∑
uv∈E(G1)
dd G1+G2 (u)dd G1+G2 (v) +
∑
uv∈E(G2)
dd G1+G2 (u)dd G1+G2 (v)
+
∑
u∈V (G1),v∈V (G2)
dd G1+G2 (u)dd G1+G2 (v)
We will find every part independently
(1)
∑
uv∈E(G1)
dd G1+G2 (u)dd G1+G2 (v) =
∑
uv∈E(G1)
(dd G1 (u) + n2)(dd G1 (v) + n2)
= DM2(G1)(1 + n2) + m1n
2
2
Int. J. Anal. Appl. 19 (1) (2021) 59
(2)
∑
uv∈E(G2)
dd G1+G2 (u)dd G1+G2 (v) =
∑
uv∈E(G2)
(dd G2 (u) + n1)(dd G2 (v) + n1)
= DM2(G2)(1 + n1) + m2n
2
1
(3)
∑
u∈V (G1),v∈V (G2)
dd G1+G2 (u)dd G1+G2 (v) = (dd G1 (u1) + n2)(dd G2 (v1) + n1) + ...
+ (dd G1 (u1) + n2)(dd G2 (vn2 ) + n1)
+ (dd G1 (u2) + n2)(dd G2 (v1) + n1) + ...
+ (dd G1 (u2) + n2)(dd G2 (vn2 ) + n1)
+ ...
+ (dd G1 (un1 ) + n2)(dd G2 (v1) + n1) + ...
+ (dd G1 (un1 ) + n2)(dd G2 (vn2 ) + n1)
= (dd G1 (u1) + n2)[
∑
v∈V (G2)
(dd G2 (v) + n1)] + ...
+ (dd G1 (un1 ) + n2)[
∑
v∈V (G2)
(dd G2 (v) + n1)]
= [
∑
u∈V (G1)
(dd G1 (u) + n2)][
∑
v∈V (G2)
(dd G2 (v) + n1)]
= [n1n2 +
∑
u∈V (G1)
dd G1 (u)][n1n2 +
∑
v∈V (G2)
dd G2 (v)]
= [n1n2 + ρ(G1)][n1n2 + ρ(G2)]
Hence,
DM2(G1 + G2) = DM2(G1)(1 + n2) + m1n
2
2 + DM2(G2)(1 + n1) + m2n
2
1
+ [n1n2 + ρ(G1)][n1n2 + ρ(G2)].
And,
DM∗1 (G1 + G2) =
∑
uv∈E(G1+G2)
[ddG1+G2 (u) + ddG1+G2 (v)]
=
∑
uv∈E(G1)
[dd G1+G2 (u) + dd G1+G2 (v)] +
∑
uv∈E(G2)
[dd G1+G2 (u) + dd G1+G2 (v)]
+
∑
u∈V (G1),v∈V (G2)
[dd G1+G2 (u) + dd G1+G2 (v)].
Int. J. Anal. Appl. 19 (1) (2021) 60
We will find every part independently
(1)
∑
uv∈E(G1)
[dd G1+G2 (u) + dd G1+G2 (v)] =
∑
uv∈E(G1)
(dd G1 (u) + n2) + (dd G1 (v) + n2)
= DM∗1 (G1) + 2m1n2,
(2)
∑
uv∈E(G2)
[dd G1+G2 (u) + dd G1+G2 (v)] =
∑
uv∈E(G2)
(dd G2 (u) + n1) + (dd G2 (v) + n1)
= DM∗1 (G2) + 2n1m2,
(3)
∑
u∈V (G1),v∈V (G2)
[dd G1+G2 (u) + dd G1+G2 (v)] = (dd G1 (u1) + n2) + (dd G2 (v1) + n1) + ...
+ (dd G1 (u1) + n2) + (dd G2 (vn2 ) + n1)
+ (dd G1 (u2) + n2) + (dd G2 (v1) + n1) + ...
+ (dd G1 (u2) + n2) + (dd G2 (vn2 ) + n1)
+ ...
+ (dd G1 (un1 ) + n2) + (dd G2 (v1) + n1) + ...
+ (dd G1 (un1 ) + n2) + (dd G2 (vn2 ) + n1)
= (dd G1 (u1) + n2)[
∑
v∈V (G2)
(dd G2 (v) + n1) + n2] + ...
+ (dd G1 (un1 ) + n2)[
∑
v∈V (G2)
(dd G2 (v) + n1) + n2]
= [
∑
v∈V (G2)
(dd G2 (v)) + n2(n1 + 1)][
∑
u∈V (G1)
(dd G1 (u)) + n1n2)]
= (ρ(G2) + n2(n1 + 1))(ρ(G1) + n1n2).
Hence,
DM∗1 (G1 + G2) = DM
∗
1 (G1) + DM
∗
1 (G2) + 2m1n2 + 2n1m2
+ (ρ(G2) + n2(n1 + 1))(ρ(G1) + n1n2).
�
Int. J. Anal. Appl. 19 (1) (2021) 61
Corollary 3.2.
DM1(G1 + G2) ≤ DM1(G1) + DM2(G2) + 2n1Γ(G1)Tm(G1)
+ 2n1Γ(G2)Tm(G2) + n1(n
2
2 + n2n1).
Corollary 3.3.
DM2(G1 + G2) ≤ DM2(G1)(1 + n2) + m1n22 + DM2(G2)(1 + n1) + m2n
2
1
+ [n1n2 + Γ(G1)Tm(G1)][n1n2 + Γ(G2)Tm(G2)].
Corollary 3.4.
DM∗1 (G1 + G2) ≤ DM
∗
1 (G1) + DM
∗
1 (G2) + 2m1n2 + 2n1m2
+ (Γ(G2)Tm(G2) + n2(n1 + 1))(Γ(G1)Tm(G1) + n1n2).
4. Some bounds of domination Zagreb indices
Theorem 4.1.
Let G be a graph of order n. Then
DM1(G) ≥ 1n (ρ(G))
2. Equality hold if and only if G is one-domination regular graph.
Proof. We have DM1(G) =
∑
v∈V (G) d
2
d(v) = d
2
d(v1) + d
2
d(v2) + ... + d
2
d(vn)
We use Cauchy-schwartz inequality on vectors (dd(v1),dd(v2), ...,dd(vn)) and (1, 1, ..., 1) to get
DM1(G).n = (d
2
d(v1),d
2
d(v2), ...,d
2
d(vn))(1
2, 12, ..., 12)
≥ (dd(v1).1 + dd(v2).1 + ... + dd(vn).1)2
= (
n∑
i=1
dd(vi))
2
= (ρ(G))2.
To prove the equality, suppose G is one domination regular graph ⇒ dd(vi) = 1 for all 1 ≤ i ≤ n. And
DM1(G) = n. Conversely if DM1(G) =
1
n
(
∑n
i=1 dd(vi))
2 ⇒ DM1(G).n = (
∑n
i=1 d(vi))
2, hence G is one
domination regular graph. �
Theorem 4.2.
Let G be a graph with n vertices. Then
DM1(G) ≤ (
∑
v∈V (G)
√
dd(v))
2.
Int. J. Anal. Appl. 19 (1) (2021) 62
Proof. DM1(G) =
∑
v∈V (G) d
2
d(v) = d
2
d(v1) + d
2
d(v2) + ... + d
2
d(vn)
As dd(v1),dd(v2), ...,dd(vn) is positive integers so, we get
DM1(G) ≤ (
∑
v∈V (G)
√
dd(v))
2.
�
Proposition 4.1.
If G be any graph such that |V (G)| = n, then
n ≤ DM1(G) ≤ n(Tm(G))2 ,
ρ(G) + n ≤ DM∗1 (G) ≤ nTm(G) + ρ(G).
Proof. see Observation 2.1. �
Theorem 4.3.
Let G be a graph such that G 6∼= K2. Then DM2(G) ≥ γ(G)Tm(G)
Equality hold if and only if G ∼= P3.
Proof. Note that dd(v) ≥ 1, so,
∑
uv∈E(G) dd(u)dd(v) ≥
∑
u∈V (G) dd(u). From Observation 2.3, we get
DM2(G) ≥ γ(G)Tm(G). �
Theorem 4.4.
Suppose G is a connected simple graph. Then
DM2(G) ≤ Γ(G)(Tm(G))2.
Proof. We have dd(v) ≤ Tm(G), so
∑
uv∈E(G) dd(u)dd(v) ≤ Tm(G)
∑
u∈V (G) dd(u).
By Observation 2.3, DM2(G) ≤ Γ(G)(Tm(G))2. �
Finally, we can generalize the definition of the domination degree of the vertex by using any subset of
vertices with property P like an independent set, independent dominating set, total dominating set, hup set,
edge dominating set, different distance set,... so on.
Definition 4.1.
Let G = (V,E) be a graph and let S be any subset of vertices with property P . Then for any vertex v, the P
set degree of the vertex v denoted by
dP (v) = |{S ⊆ V (G) : S has property P and v ∈ S}|.
Int. J. Anal. Appl. 19 (1) (2021) 63
And we can define the Zagreb and Forgotten indices as following
PM1(G) =
∑
v∈V (G)
d2P (v) ,
PM∗1 (G) =
∑
uv∈E(G)
dP (u) + dP (v) ,
PM2(G) =
∑
uv∈E(G)
dP (u)dP (v) ,
PF(G) =
∑
v∈V (G)
d3P (v) ,
PF∗(G) =
∑
uv∈E(G)
d2P (u) + d
2
P (v) .
5. Conclusion
In this research work, we define new topological indices based on the minimal dominating sets. The
authors are working now in some of the other types of topological indices by replacing the standers degree
of the vertex by the domination degree of the vertex.
Finally we have the following open problems for research:
(1) Is there any relation between the normal degree and domination degree of the graph.
(2) What is the necessary and sufficient conditions in a graph to become domination regular graph.
(3) What is the natural relation between the domination indices.
Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication
of this paper.
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1. Introduction
2. Domination Degree in Graphs
3. Domination Zagreb Indices of a graph
4. Some bounds of domination Zagreb indices
5. Conclusion
References