Properties of Graph Based on Divisor-Euler Functions

Divisor function 𝐷𝐷 ( 𝑛𝑛 ) gives the residues of 𝑛𝑛 which divide it. A function denoted by 𝜏𝜏 ( 𝑛𝑛 ) counts the total possible divisors of 𝑛𝑛 and 𝜙𝜙 gives the list of co-prime integers to 𝑛𝑛 . Many graphs had been constructed over these arithmetic functions. Using 𝐷𝐷 ( 𝑛𝑛 ) and 𝜙𝜙 ( 𝑛𝑛 ) , a well known graph named as divisor Euler function graph has been constructed. In this paper, we use divisor function and anti Euler function 𝜙𝜙 ′ . We label the symbol 𝜙𝜙 𝑐𝑐 ( 𝑛𝑛 ) to count those residues of 𝑛𝑛 which are not co-prime to 𝑛𝑛 . By using these functions, we find a new graph, called divisor anti-Euler function graph (DAEFG), denoted as 𝐺𝐺 ( 𝑑𝑑 ( 𝜙𝜙 𝑐𝑐 ( 𝑛𝑛 )) . Let 𝐺𝐺 ( 𝑑𝑑 ( 𝜙𝜙 𝑐𝑐 ( 𝑛𝑛 )) = ( 𝕍𝕍 , 𝔼𝔼 ) be a DAEFG, where 𝕍𝕍 = { 𝑑𝑑


INTRODUCTION
The systematic use of number theoretic functions in combinatorial mathematics is an interesting and useful task nowadays.Recently, the study of graphs based on number theoretic functions has become much more motivating.In our work, we use three number theoretic functions and define a new class of graph based on these functions.We familiarize and study the structures of such graphs.Such construction of graphs based on number theory leads to many new useful results.This area of mathematics has a wide range of applications in chemistry, computer science, navigation, robot science and engineering as well.The notion of divisor graph was first introduced by Singh and Santhosh in 2000.In 2015, K. Kannan et al. [1] presented a graph of number theocratic function namely, the divisor function graph.He suggested an idea for constructing graphs on divisors of integers by taking as vertices.We denote the set () = �  :   | ,   ≤ � as the set of positive divisors of .In his contribution, both vertices and edges were based on ().Shanmugavelan constructed a graph, namely Euler's phi function graph based on Euler's phi function [2].He constructed this graph by taking a set of nodes and a set of edges based on this function ().Using these co-prime residues, an idea of prime labeling has been introduced and investigated in [3].Over  vertices, the formula for finding the maximal number of edges in this type of labeling has been established as Σ() in [3].The well-known Cayley graph associated with the totient function, known as the Cayley totient graph [4], contains residues modulo  namely {0,1,2, … ,  − 1} with the edge set {(, )/ −  ∈ }.It is denoted by �(  )�, where  denotes all positive integers which are less than  and co-prime to .In the following paragraph, a few definitions are addressed to make this study self-contained.
Metric dimension is an attractive parameter in graph theory.The idea of resolving set for a connected graph was firstly first used by Slater [5] and [6], where he termed it as a locating set.He referred to the least locating set as a reference set and its cardinality as the Metric dimension, which has a wide range of applications in many fields of chemistry computer science physics and robot science.The distance between any two vertices is denoted as (, ) which gives the minimum number of edges needed to transverse to reach from  to .Let  be a graph with one component i.e., a connected graph, a node  resolves a pair of vertices  and  of () if (, ) ≠ (, ).If a subset ℜ ⊆  resolves the whole set of nodes, then ℜ is called a resolving set (RS).General results on MD were discussed in [12].Eccentricity of a particular vertex  is defined as the maximum distance between any of the vertices of the graph with  that is ecc () = max{(, );  lies in ()}.Many useful results regarding FMD and LFMD were discussed in [7] and [8], respectively.Results regarding algorithm and graph labeling were discussed in [9][10][11].Many results on digraphs and their labeling based on number theory are discussed in [12][13][14][15].Results on upper bound sequence of networks and RN's via Lambert type Map can be seen in [16][17][18].Useful results on further graph theoretic applications are widely discussed in [19][20][21].Research on degree and connection-based Zagreb indices of the network is astonishing nowadays.
Fruitful results of such newly defined degree based topological invariants of the M-polynomial, tadepol graph are discussed in [22] and [23].Computation of entropy measures and valency-based indices of networks are discussed in [24] and [25].The results of connection based such indices of networks are given in [26][27][28].
A loop (or a fixed point) in a graph is a vertex that is adjacent to itself, and it is called an isolated vertex if it is not adjacent to any other vertex.The vertices  1 ,  2 , . .,   will form a cycle of length  if  1 is adjacent to  2 ,  2 is adjacent to  3 , and so on till   is adjacent to  1 .A maximal connected subgraph is termed as a component.The number of vertices adjacent to a vertex  is called its degree.A graph is complete if all vertices are adjacent to the rest of all vertices.A graph is termed as bipartite if its set of vertices can be partitioned into two disjoint sets such that any two vertices in a set are not adjacent to each other.A graph is regular if all vertices have same degree.A graph is said to be planner if it can be drawn on a plane such that no two edges intersect each other except the end points.A graph is termed as Hamiltonian if it has a cycle through all vertices and visits each vertex exactly one time.While a graph is Eulerian if each of its vertex has an even degree.The chromatic number is the smallest number of colors that can be assigned to its vertices such that no two same colors are adjacent or no two adjacent vertices have same color and the minimum number of colors such that no two adjacent edges have same color.Lastly, the order of a maximal complete subgraph is called a clique of that graph.Now, we recall a few well-known definitions and results from number theory which will be used in the sequel to make this paper self-contained.The following definitions can be found in [7].Definition 1.6.[4] The number of integers from the set {1,2, … ,  − 1} which are not relatively prime to  is the function  ′ (). ′ () denotes the numbers that are less than or equal to  and non-prime to .Since () =  − 1 for  be any prime but  ′ () = 1, for  = 5,  ′ () = 1, and  ′ (8) = 4, and these numbers are 2, 4, 6 and 8 as these are all non-prime to , i.e., (8,20) Theorem 1.2.[4] If  is any prime, then (  ) =  −1 ( − 1),  ≥ 1.  Proof: The proof is simple.Since node 1 is isolated which clearly depicts that no other node can have an edge with node 1.So, there are total () − 1 nodes in the connected part of graph.As by definition graph is simple so there is no loop.So, there are () − 2 total number of possible edges incident with node .Hence, it is the only node with the largest degree which is () − 2. Since SAEFG is a simple graph so there is no loop at any vertex in particular there is no loop at vertex  itself.Since there are () possible vertices of DAFTG.Also, each   ≠ 1 for each i is adjacent to n (possibly).These   are () − 2 in number (excluding 1 and n ), so d must have degree () − 2.
Theorem 2.1.DAEFG is connected and 1 is the only isolated vertex in it.Proof: Let  1 ,  2 , … … ,   all the possible divisors of any positive integer , where   = .Since 1 divides each integer, then  ∈ .By definition (1,   ) = 1, So 1 cannot join any of the node which means that node 1 is isolated.Now, its only need to show that  is connected.Since each   divide  and which is not co prime to  i.e., (  , ) ≥ 1.So, clearly there is an edge between each   to .Hence,  is connected.
Proof.The proof is simple.Since, the node 1 is isolated which clearly depicts that no other node can have an edge with node 1.So, there are total () − 1 nodes in the connected part of graph.As by definition graph is simple so there is no loop.So, there are () − 2 total number of possible edges incident with node .Hence, it is the only node with the largest degree which is () − 2. Since, SAEFG is a simple graph so there is no loop at any vertex in particular there is no loop at vertex  itself.Since there are () possible vertices of DAFTG.Also, each   ≠ 1 for each i is adjacent to n (possibly).These   are () − 2 in no.(excluding 1 and n ), so d must have degree () − 2.
Proof: Since, {1,2, 2 2 , 2 3 , … , 2  } be the set of all possible nodes.As deg (1) = 0 so node 1 is not adjacent with any other node.By definition of  there are  + 1 total possible divisors.By excluding node 1 there are  total possible nodes.Since node 2 is even and each node is of the power of 2 which is again an even number so each node is adjacent to the node 2  which is the required .Since the graph is simple so there is neither a loop nor a multi-edge.As each node is adjacent to every other node except itself and the node 1. Hence there are  − 1 total possible edges for each node, which is the property of a complete graph by definition.
Lemma 2.1.The graph  is bipartite for  be the product of two distinct primes.
Proof: The proof can be viewed using its definition.For  =  then  = {1, , , } be the set of nodes.Since, node 1 is isolated and we take the set of nodes excluding node 1 for the connected part of graph.So, the set of nodes can be split into two disjoint sets as  and  such that  = {, } and  = {} with vertex 1 as isolated.As (, ) ≥ 1 and (, ) ≥ 1 but (, ) = 1, so there are edges as  −  and  − , where there is no edge between  and .

Corollary 2.4.
There does not exist any cycle for  for  = 2  .

Proof:
The proof is a simple consequence Corollary 2.2.Proof: Let  = 2  for  then by definition of ,  = {1,2, 2 2 , 2 3 , … , 2  } be the set of nodes.Also, �1, 2  � = 1 which gives that 1 is not adjacent with any of the nodes among the set of nodes, i.e., 1 is isolated.Since, 2 is even and each of its power is again even i.e., �2  , 2  � ≥ 1 also 2  is again even, which gives that each node is adjacent to every other node which yield that  is a complete graph.Also, degree of each node is same in the connected component of .By using the result of completeness  is also regular for  = 2  .Let  be DAEFG, to show that  is regular, each of its vertices should have the same degree.
Theorem 2.7. is planar except for  = 2  ,  ≥ 5. Proof: For n ≥ 3, (i) if  = , () = 2, and () = {1, }.Using definition there is no edge between node 1 and the p which gives a null graph.(ii) Secondly, if it is not prime then it is a composite number, which can be even or odd.(i)Suppose that it is even then, () = 2 m i.e deg (1) = 0 which shows that node 1 is not adjacent with any of the nodes so the max possible degree of other nodes can be deg (  ) = () − 1∀  ∈ (), which is not even, which contradicts the necessity condition for elerianity of a graph to have even degree, which gives that  is not eulerian.(ii)On the other hand suppose that  is odd, then there are following possibilities.(i) If () = even using above statement, the result if obvious.(ii) But, if () = , we have all nodes of degree odd in number and no such trail passing via all edges, which is the required result.

Proof:
The proof is a simple consequence of previous theorem that  is k colorable for   as exactly k colors are needed to color its set of nodes.Also, it the least number of possible such coloring.Proof.By using the definition of , () = {1, }.Also, gcd (1, ) = 1 and  ≈ .In this case, exactly one color is sufficient to give both of the nodes in order to color .
Proof: Since, the graph has exactly 2 components with 1 as isolated and 1, …,  are the  possible divisors of , Since, () − {1} vertices constitutes  , with even cardinality.As all prime powers are not relatively prime to each other so they are adjacent via edges, so there is no other option but to assign them with a different color.By assigning distinct color to each edge, () − {1} colors are needed, which is the required result.
Proof: Since,  ′ be the least possible coloring assigned to edges and () is constructed using edges of  using nodes and  is the least possible coloring assigned to nodes.Thus, it can be easily seen that  ′ () = (()).

CONCLUSION
In this work, we have studied the structure of Divisor Anti Euler Function graph DAEFG.We computed its order, degree of nodes, number of components, length of cycle, its subgraphs and other graph theoretic properties.Furthermore, we found chromatic number, chromatic index, Hamiltonicity, Eulerianity, regularity and bipartiteness.In future, we find

Definition 1 . 1 .Definition 1 . 5 .
[4] Arithmetic functions are those functions, which are defined for all positive integers, such as Divisor function (), Euler Phi function (), Tau Function (), Sigma function () and Anti-Euler function  ′ etc. Definition 1.2.[4] Divisor function () is the set of those numbers which are less then or equal to  and which divides .For example, for  = 10, () = {1,2,5,10} Definition 1.3.[2] Divisor Euler function graph  = (, )is a graph in which set of nodes are based on divisor function and set of edges is based on Euler phi function and any two nodes are adjacent if these are co-prime to each other.Definition 1.4.[2] Divisor-not prime function graph is a graph in which set of nodes is based on divisor function and any two nodes are adjacent if these are not prime to each other.It is also termed as divisor anti Euler function graph.[4] Divisor Euler function graph denoted by () is a graph in which the number of integers from the set {1,2, … ,  − 1} are relatively prime to , i.e., (n) = 1.