Murat Cancan

Abstract:Topological indices enable to gather information for the underlying topology of chemical structures and networks.Novel harmonic indices have been defined recently.All degree based topological indices are defined by using the classical degree concept.Recently two novel degree concept have been defined in graph theory:ve-degree and evdegree.Ve-degree Zagreb indices have been defined by using ve-degree concept.The prediction power of the ve-degree Zagreb indices is stronger than the classical Zagreb indices.Dominating oxide,silicate and oxygen networks are important network models in view of chemistry,physics and information science.Physical and mathematical properties of dominating oxide,silicate and oxygen networks have been considerably studied in graph theory and network theory.Topological properties of the dominating oxide,silicate and oxygen networks have been intensively investigated for the last few years period.In this study we examined,the first,the fifth harmonic and ev-degree topological indices of dominating oxide (DOX),regular triangulene oxide network (RTOX)and dominating silicate network (DSL).

Keywords:Dominating oxide network,dominating silicate network,ev-degree topological indices,harmonic indices,regular triangulene oxide network.

1 Introduction

Graph theory has many applications for science,technology and social sciences.Graph theory enables suitable toys to researches to model real world problems.Chemical graph theory is one of the branch of graph theory.Chemical graph theory is considered the intersection of graph theory,chemistry and information science.In chemistry,pharmacology,medicine and physics molecular graphs has been used to model atomic and molecular substances.Topological indices have been derived from the molecular graphs of chemical compounds.Topological indices are important tools to analyze the underlying topology of networks.Many topological indices have been used to understand and to investigate mathematical properties of real world network models.Topological indices enable to gather information for the underlying topology of chemical structures and networks.Zagreb and Randic indices are the most used indices among the all topological indices in this regard [Gutman and Trinajsti? (1972);Randic (1975)].Harmonic index was defined by Zhong [Zhong (2012)].Novel harmonic indices have been defined by Ediz et al.[Ediz,Farahani and Imran (2017)].All degree based topological indices are defined by using the classical degree concept.Recently two novel degree concept have been defined in graph theory:ve-degree and ev-degree [Chellali,Haynes,Hedetniemi et al.(2017)].On mathematical properties of these novel concepts we referred the interested reader to the reference [Horoldagva,Das,Selenge (2019)].Ev-degree and Ve-degree Zagreb indices and Ev-degree and Ve-degree Randic indices have been defined by using ve-degree concept in Ediz et al.[Ediz (2017a,2017b,2018);?ahin and Ediz (2018)].The prediction power of the ve-degree Zagreb indices is stronger than the classical Zagreb indices.Dominating oxide network is important toll in view of chemistry,physics and information science.Physical and mathematical properties of dominating oxide network have been considerably studied in graph theory and network theory.Topological properties of the dominating oxide network have been intensively investigated for the last five years period see in Arockiaraj et al.[Arockiaraj,Kavitha,Balasubramanian et al.(2018);Baig,Imran and Ali (2015);Gao and Siddiqui (2017);Sarkar,De,Cangül et al.(2019);Simonraj and George (2013)].

The ve-degree topological properties of dominating oxide network have been investigated by [Kulli (2018a));Kulli (2018b)].As a continuation of these studies,in this study we examined,ev-degree Zagreb,ev-degree Randic and the first,the fifth harmonic topological indices of DOX,RTOX and DSL networks.

2 Preliminaries

In this section we give some basic and preliminary concepts which we shall use later.A graphG=(V,E)consists of two nonempty setsVand 2-element subsets ofVnamelyE.The elements ofVare called vertices and the elements ofEare called edges.For a vertexv,deg(v)show the number of edges that incident tov.The set of all vertices which adjacent to v is called the open neighborhood ofvand denoted byN(v).If we add the vertexvtoN(v),then we get the closed neighborhood ofv,N[v].For the vertices u and v,d(u,v)denotes the distance between u and v which means that minimum number of edges between u and v.The largest distance from the vertex v to any other vertex u called the eccentricity of v and denoted by ev.

LetGbe a simple connected graphG=(V,E).Harmonic indices may be defined as;

where Quis a unique parameter which is acquired from the vertex u ∈V(G).

Definition 1 (First Harmonic Index)The first kind of this Harmonic indices was studied by [Zhong,L.(2012)] by considering Quto be the degree of the vertexu:

Definition 2 (Second Harmonic Index)The second kind of this class can be defined by considering Quto be the number nuof vertices ofGlying closer to the vertexuthan to the vertexvfor the edgeuvof the graphG:

Definition 3 (Third Harmonic Index)The third type of this class can be defined by considering Quto be the number muof edges ofGlying closer to the vertexuthan to the vertexvfor the edgeuvof the graphG:

Definition 4 (Fourth Harmonic Index)The fourth type of this class can be defined by considering Quto be the eccentricity of the vertexu:

Definiton 5 (Fifth Harmonic Index)The fifth type of this class can be defined by considering Quto be the Su=∑v∈N(u)dv:

Definition 6 (Sixth Harmonic Index)And the sixth type of this class can be defined by considering Quto be the Mu=∏v∈N(u)dv:

And now we give the definitions of ev-degree concept which were given by Chellali et al.[Chellali,Haynes,Hedetniemi et al.(2017)].

Definition 7(ve-degree)LetGbe a connected simple graph andv∈V(G).The ve-degree of the vertexv,degve(v),equals the number of different edges that incident to any vertex from the closed neighborhood ofv.

We also can restate the Definition 7 as follows:LetGbe a connected simple graph andv∈V(G).The ve-degree of the vertexvis the number of different edges between the other vertices with a maximum distance of two from the vertexv.

Definition 8(ev-degree)LetGbe a connected graph ande=uv∈E(G).The ev-degree of the edgee,degev(e),equals the number of vertices of the union of the closed neighborhoods ofuandv.

The authors in Chellali et al.[Chellali,Haynes,Hedetniemi et al.(2017)] also can give the Definition 8 as follows:LetGbe a connected graph ande=uv∈E(G).The ev-degree of the edgee,deguv(e)=degu+degv-ne,wherenemeans the number of triangles in which the edgeelies in.

Definition 9 (ev-degree Zagreb index)LetGbe a connected graph ande=uv∈E(G).The ev-degree Zagreb index of the graphGdefined as;

Definition 10 (the first ve-degree Zagreb alpha index)LetGbe a connected graph andv∈V(G).The first ve-degree Zagreb alpha index of the graphGdefined as;

Definition 11 (the first ve-degree Zagreb beta index)LetGbe a connected graph anduv∈E(G).The first ve-degree Zagreb beta index of the graphGdefined as;

Definition 12 (the second ve-degree Zagreb index)LetGbe a connected graph anduv∈E(G).The second ve-degree Zagreb index of the graphGdefined as;

Definition 13 (ve-degree Randic index)LetGbe a connected graph anduv∈E(G).The ve-degree Randic index of the graphGdefined as;

Definition 14 (ev-degree Randic index))LetGbe a connected graph ande=uv∈E(G).The ev-degree Randic index of the graphGdefined as;

After these definitions,we calculate the first and the fifth harmonic and ev-degree indices for the DOX,DSL and RTOX networks in the following sections.

3 Harmonic,ev-degree Zagreb and ev-degree Randic indices of dominating oxide network (DOX)

The structure of a DOX network is depicted in Fig.1.Before we calculate the ev-degree and the first and the fifth harmonic topological indices of DOX network we have to determine the ev-degrees of end vertices of the all edges for an arbitrary DOX network.

Figure 1:A dominating oxide DOX(2)network model

Observe that from Fig.1,every edge of dominating oxide network lies in only one triangle.In this regard,the ev-degree of the each edge of dominating oxide network is equal to the degree of sum of end vertices minus one.The following Tab.1,shows the partition of the edges with respect to their ev-degree of end vertices for an arbitrary DOX network.

Table 1:The ev-degrees of the end vertices of edges for DOX networks

Table 2:The sum degrees of the end vertices of edges for DOX networks

And now,we begin to compute ev-degree and harmonic topological indices for DOX networks.

Theorem 1The ev-degree Zagreb index of an arbitrary dominating oxide network is equal to 2646n2-3222n+1170.

ProofFrom the definition of the ev-degree Zagreb index and Tab.1,we can directly write;

Theorem 2The ev-degree Randic index of an arbitrary dominating oxide network is equal to

ProofFrom the definition of the ev-degree Randic index and Tab.1,we can directly write;

Theorem 3The first harmonic index of an arbitrary dominating oxide network is equal to

ProofFrom the definition of the first harmonic index and Tab.1,we can directly write;

Theorem 4The fifth harmonic index of an arbitrary dominating oxide network is equal to

ProofFrom the definition of the first harmonic index and Tab.2,we can directly write;

4 Harmonic,ev-degree Zagreb and ev-degree Randic indices of regular triangulene oxide network (RTOX)

The structure of a RTOX network is depicted in Fig.2.Before we calculate the ev-degree and the first and the fifth harmonic topological indices of RTOX network we have to determine the ev-degrees of the all edges for an arbitrary RTOX network.

Figure 2:Regular triangulate oxide network RTOX(5)

Table 3:Edge partition of the regular triangulene oxide network RTOX(n)(n ≥3)

The following Tab.4,shows the partition of the edges with respect to their sum degree of end vertices an arbitrary RTOX network.

Table 4:The sum degrees of the end vertices of edges for RTOX(n)(n ≥3)networks

And now,we begin to compute ev-degree and harmonic topological indices for RTOX networks.

Theorem 5The ev-degree Zagreb index of an arbitrary triangulene oxide network (RTOX)is equal to 247n2+150n-80.

ProofFrom the definition of the ev-degree Zagreb index and Tab.3,we can directly write;

Theorem 6The ev-degree Randic index of an arbitrary triangulene oxide network (RTOX)is equal to

ProofFrom the definition of the ev-degree Randic index and Tab.3,we can directly write;

Theorem 7The first harmonic index of an arbitrary triangulene oxide network (RTOX)is equal to

ProofFrom the definition of the first harmonic index and Tab.3,we can directly write;

Theorem 8The fifth harmonic index of an arbitrary triangulene oxide network (RTOX)is equal to

ProofFrom the definition of the first harmonic index and Tab.4,we can directly write;

5 Harmonic,ev-degree Zagreb and ev-degree Randic indices of dominating silicate network (DSL)

The structure of a DSL network is depicted in Fig.3.Before we calculate the ve-degree topological indices of DSL network we have to determine the ev-degrees of all the edges for an arbitrary DSL network.The following Tab.5,shows the partition of the edges with respect to their sum degree of end vertices of an arbitrary DSL network.

Figure 3:Dominating silicate network DSL(2)

Table 5:Edge partition of the dominating silicate network DSL(n)

Table 6:The sum degrees of the end vertices of edges for DSL(n)networks

Theorem 9The ev-degree Zagreb index of an arbitrary dominating silicate network (DSL)is equal to 9180n2-11304n+4122.

ProofFrom the definition of the ev-degree Zagreb index and Tab.5,we can directly write;

Theorem 10The ev-degree Randic index of an arbitrary dominating silicate network (DSL)is equal to

ProofFrom the definition of the ev-degree Randic index and Tab.5,we can directly write;

Theorem 11The first harmonic index of an arbitrary dominating oxide network is equal to

ProofFrom the definition of the first harmonic index and Tab.5,we can directly write;

Theorem 12The fifth harmonic index of an arbitrary dominating silicate network (DSL)is equal to

ProofFrom the definition of the first harmonic index and Tab.6,we can directly write;

6 Conclusion

The ev-degree topological indices have been defined recently.We examined the ev-degree Zagreb indices and ev-degree Randic indices of dominating oxide,silicate and oxygen networks.Also,we calculated the first,the fifth and the sixth harmonic indices of the same networks.The mathematical properties of ev-degree and harmonic indices have not been investigated so far.The mathematical properties of ev-degree and harmonic indices are worth to study for future researches.