A Polynomial Representation and a Unique Code of a Simple Undirected Graph

Abstract

We introduce a representation of simple undirected graphs in terms of polynomials and obtain a unique code for a simple undirected graph.

1. Introduction

Let 𝑀 be the set of all positive integers greater than 1. Let 𝑛∈𝑀 and 𝑉(𝑛) be the set of all divisors of 𝑛, greater than 1. Define a simple undirected graph 𝐺(𝑛)=(𝑉,𝐸) with the vertex set 𝑉=𝑉(𝑛) and any two distinct vertices 𝑎,𝑏∈𝑉 are adjacent if and only if gcd(𝑎,𝑏)>1. From an observation in [1], it follows that any simple undirected graph is isomorphic to an induced subgraph of 𝐺(𝑛) for some 𝑛∈𝑀. For the sake of completeness and further use of the construction we provide a sketch of the proof below. Throughout the paper by a graph, we mean a simple undirected graph.

Theorem 1. Let 𝐺 be a graph. Then 𝐺 is isomorphic to an induced subgraph of 𝐺(𝑛) for some 𝑛∈𝑀.

Proof. Let 𝐺=(𝑉,𝐸) be a graph. Let {𝐶₁,𝐶₂,…,𝐶_𝑘} be the set of all maximal cliques of 𝐺. For 𝑖=1,2,…,𝑘, let 𝑝_𝑖 be the 𝑖th prime. For each 𝑣∈𝑉, define 𝑠₁(𝑣)=∏{𝑝_𝑗|𝑣∈𝐶_𝑗}. Now in order to make the values of 𝑠₁(𝑣) distinct for distinct vertices, we modify 𝑠₁(𝑣) by using different powers of primes 𝑝_𝑗, if required. For each 𝑣∈𝑉, let 𝑠(𝑣) be the modified value of 𝑠₁(𝑣). Let 𝑛 be the least common multiple of {𝑠(𝑣)|𝑣∈𝑉}. Now it is clear that for any 𝑢,𝑣∈𝑉, 𝑢isadjacentto𝑣in𝐺⟺𝑢,𝑣∈𝐶_𝑖forsome𝑖∈{1,2,…,𝑘}⟺𝑝_𝑖(𝑢)(𝑣){}(𝑠(𝑢)(𝑣(𝑢)𝑠(𝑣)(𝑛).isafactorofboth𝑠and𝑠forsome𝑖∈1,2,…,𝑘⟺gcd,𝑠))>1⟺𝑠isadjacenttoin𝐺 (1)  Thus, 𝐺 is isomorphic to the subgraph of 𝐺(𝑛) induced by the set {𝑠(𝑣)|𝑣∈𝑉} of vertices of 𝐺(𝑛).

Example 2. Consider the graph 𝐺=(𝑉,𝐸) in Figure 1. The maximal cliques of 𝐺 are 𝐶₁={𝑣₁,𝑣₂,𝑣₃,𝑣₄}, 𝐶₂={𝑣₂,𝑣₃,𝑣₄,𝑣₅,𝑣₆}, 𝐶₃={𝑣₇,𝑣₈,𝑣₁₀}, 𝐶₄={𝑣₉,𝑣₁₀}, and 𝐶₅={𝑣₁₀,𝑣₁₁}. We assign the 𝑖th prime to the clique 𝐶_𝑖 for 𝑖=1,2,…,5. Then for each 𝑣∈𝑉, we compute 𝑠₁(𝑣) and 𝑠(𝑣) as in the proof of Theorem 1 (Table 1).
So 𝐺 is isomorphic to the subgraph of 𝐺(𝑛) induced by the set {2,3,5,6,7,9,11,12,18,25,385} of vertices of 𝐺(𝑛), where 𝑛=2²⋅3²⋅5²⋅7⋅11=69300.

Table 1 

Figure 1 

The graph 𝐺 in Example 2.

2. Main Results

It is important to note that instead of taking all maximal cliques of 𝐺 in Theorem 1, it is sufficient to consider a set of cliques of 𝐺 which covers both vertices and edges of 𝐺.

Definition 3. Let 𝐺=(𝑉,𝐸) be a graph. A set 𝑆 of cliques of 𝐺 is called a total clique covering of 𝐺 if 𝑆 covers both 𝑉 and 𝐸. The minimum size of a total clique covering of 𝐺 is called the total clique covering number of 𝐺. We denote it by 𝜃_𝑡(𝐺).

On the other hand, given a finite sequence of positive integers, one can construct a simple undirected graph as follows.

Definition 4. Let 𝜎=(𝑎₁,𝑎₂,…,𝑎_𝑚) be a finite sequence of positive integers. Then corresponding to this sequence, define a graph 𝐺[𝜎]=(𝑉,𝐸), where 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑚}, 𝑣_𝑖 corresponds to 𝑎_𝑖 (𝑎_𝑖 is called the label of 𝑣_𝑖) for 𝑖=1,2,…,𝑚, and 𝑣_𝑖𝑣_𝑗∈𝐸 if and only if 𝑖≠𝑗 and gcd(𝑎_𝑖,𝑎_𝑗)>1. The graph 𝐺[𝜎] is said to be realized by the sequence 𝜎.

Now by Theorem 1, every graph can be realized by some sequence of positive integers. Also in Definition 4, it is sufficient to take the entries 𝑎_𝑖 of 𝜎 square-free if 𝑎_𝑖>1. Thus, for convenience, we specify the sequence in the following manner.

Definition 5. Let 𝐺=(𝑉,𝐸) be a graph. Let |𝑉|>1, and let the graph 𝐺₁ be obtained from 𝐺 by deleting isolated vertices of 𝐺, if there are any. Let 𝜎₁ be a finite nondecreasing sequence of square-free positive integers greater than 1 such that 𝐺₁≅𝐺[𝜎₁]. If 𝐺 has 𝑠 isolated vertices, then we prefix 𝑠 number of 1’s in 𝜎₁ to obtain the sequence 𝜎. For 𝑠=0, 𝜎=𝜎₁. If 𝐺 is a graph with a single vertex, then 𝜎=(1). Then 𝐺≅𝐺[𝜎] and the sequence 𝜎 is called a coding sequence of 𝐺. The entries of 𝜎, which are greater than 1, are called nontrivial. Let 𝜆(𝜎) be the least common multiple of all nontrivial entries of 𝜎.

Lemma 6. Let 𝜎=(𝜆₁,𝜆₂,…,𝜆_𝑚) be a coding sequence of a graph 𝐺=(𝑉,𝐸). Let 𝑉={𝑣₁,𝑣₂,…,𝑣_𝑚}, where 𝑣_𝑖 corresponds to 𝜆_𝑖 for all 𝑖=1,2,…,𝑚 (one writes 𝜆_𝑖=𝜇(𝑣_𝑖)). Let {𝑝₁,𝑝₂,…,𝑝_𝑘} be the set of all distinct prime factors of 𝜆(𝜎). Suppose 𝐺 has 𝑠⩾0 isolated vertices. Define 𝑆_𝑗={𝑣_𝑗} for 𝑗=1,2,…,𝑠 and for each 𝑗=1,2,…,𝑘, define 𝑆_𝑠+𝑗={𝑣_𝑖∈𝑉|𝑝_𝑗𝑑𝑖𝑣𝑖𝑑𝑒𝑠𝜇(𝑣_𝑖)=𝜆_𝑖}. Then 𝑆={𝑆_𝑗|𝑗=1,2,…,𝑠+𝑘} is a total clique covering of 𝐺 such that 𝑣∈𝑆_𝑖1∩𝑆_𝑖2∩⋯∩𝑆_𝑖𝑟(𝑖_𝑡>𝑠, 𝑡=1,2,…,𝑟) and 𝑣∉𝑆_𝑗 for all 𝑗∈{𝑠+1,𝑠+2,…,𝑠+𝑘}⧵{𝑖₁,𝑖₂,…,𝑖_𝑟} if and only if 𝜇(𝑣)=𝑝_𝑖1𝑝_𝑖2⋯𝑝_𝑖𝑟 and for each 𝑗=1,2,…,𝑠, 𝑣∈𝑆_𝑗 if and only if 𝜇(𝑣)=1.

Proof. We first note that 𝑆_𝑗 is a clique containing only the vertex 𝑣_𝑗 for 𝑗=1,2,…,𝑠. Now for each 𝑗∈{1,2,…,𝑘}, 𝑝_𝑗 divides 𝜆_𝑖 for some 𝑖∈{𝑠+1,𝑠+2,…,𝑚} as 𝑝_𝑗 divides 𝜆(𝜎). Thus, 𝑣_𝑖∈𝑆_𝑠+𝑗. So 𝑆_𝑗≠∅, for all 𝑗=1,2,…𝑠+𝑘. Also for any two vertices 𝑣_𝑟,𝑣_𝑡∈𝑆_𝑠+𝑗, 𝑝_𝑗 divides both 𝜆_𝑟 and 𝜆_𝑡. So 𝑣_𝑟 and 𝑣_𝑡 are adjacent in 𝐺. Thus, each 𝑆_𝑗 is a clique of 𝐺. Again since {𝑝₁,𝑝₂,…,𝑝_𝑘} is the set of all prime factors of 𝜆(𝜎), each vertex 𝑣∈𝑉 belongs to 𝑆_𝑗 for some 𝑗∈{1,2,…,𝑠+𝑘}. So 𝑆 covers 𝑉. Moreover, two vertices 𝑣_𝑖 and 𝑣_𝑗 are adjacent if and only if gcd(𝜆_𝑖,𝜆_𝑗)>1. So 𝜆_𝑖 and 𝜆_𝑗 must have at least one common prime factor, say, 𝑝_𝑟 and hence 𝑣_𝑖 and 𝑣_𝑗 both lie in 𝑆_𝑠+𝑟. So 𝑆 also covers 𝐸 and hence 𝑆 is a total clique covering of 𝐺. Now 𝑣∈𝑆_𝑖1∩𝑆_𝑖2∩⋯∩𝑆_𝑖𝑟 (𝑖_𝑡>𝑠, 𝑡=1,2,…,𝑟) and 𝑣∉𝑆_𝑗 for all 𝑗∈{𝑠+1,𝑠+2,…,𝑠+𝑘}⧵{𝑖₁,𝑖₂,…,𝑖_𝑟} if and only if 𝑝_𝑖1𝑝_𝑖2⋯𝑝_𝑖𝑟 divides 𝜇(𝑣) but 𝑝_𝑗 does not divide 𝜇(𝑣) for all 𝑗∈{𝑠+1,𝑠+2,…,𝑠+𝑘}⧵{𝑖₁,𝑖₂,…,𝑖_𝑟}. Now since nontrivial entries of 𝜎 are square-free integers, 𝜇(𝑣) is a product of distinct primes and hence 𝜇(𝑣)=𝑝_𝑖1𝑝_𝑖2⋯𝑝_𝑖𝑟. By Definition 5, for each 𝑗=1,2,…,𝑠, 𝑣∈𝑆_𝑗 if and only if 𝜇(𝑣)=1.

Definition 7. Let 𝜎=(𝜆₁,𝜆₂,…,𝜆_𝑚) be a coding sequence of a graph 𝐺. Then the total clique covering 𝑆 of 𝐺 as defined in Lemma 6 is called the total clique covering of 𝐺corresponding to the sequence 𝜎 and is denoted by 𝑆_𝜎.

Lemma 8. Let 𝐺=(𝑉,𝐸) be a graph, and let 𝑆 be a total clique covering of 𝐺. Then there exists a coding sequence 𝜎 of 𝐺 such that 𝑆=𝑆_𝜎.

Proof. Let 𝑆={𝐶₁,𝐶₂,…,𝐶_𝑠+𝑘}, where |𝐶_𝑖|=1 for 𝑖=1,2,…,𝑠, 𝑠⩾0. Let 𝑝_𝑖 denote the 𝑖th prime. We define a map 𝑓∶𝑉→ℕ by 𝑓(𝑣)=1 if 𝑣∈𝐶₁∪𝐶₂∪⋯∪𝐶_𝑠 and 𝑓(𝑣)=𝑝_𝑖1𝑝_𝑖2⋯𝑝_𝑖𝑟 if 𝑣∈𝐶_𝑠+𝑖1∩𝐶_𝑠+𝑖2∩⋯∩𝐶_{𝑠+𝑖𝑟}(where 1≤𝑟≤𝑘) and 𝑣∉𝐶_𝑗 for all 𝑗∈{𝑠+1,𝑠+2,…,𝑠+𝑘}⧵{𝑠+𝑖₁,𝑠+𝑖₂,…,𝑠+𝑖_𝑟}. Now 𝑆 being a total clique covering, every vertex is assigned a label in this way. Also note that two vertices 𝑢, 𝑣 are adjacent in 𝐺 if and only if they lie in some 𝐶_𝑠+𝑗∈𝑆 and consequently, if and only if 𝑓(𝑢) and 𝑓(𝑣) have a common prime factor 𝑝_𝑗. Let us arrange the vertices of 𝐺 according to the nondecreasing order of {𝑓(𝑣)|𝑣∈𝑉} and define the sequence 𝜎 by 𝜎=(𝑓(𝑣₁),𝑓(𝑣₂),…,𝑓(𝑣_𝑚)), where 𝑓(𝑣₁)≤𝑓(𝑣₂)≤⋯≤𝑓(𝑣_𝑚). Then clearly 𝜎 is a coding sequence of 𝐺 and 𝑆=𝑆_𝜎.

Theorem 9. Let 𝐺 be a graph with 𝑠⩾0 isolated vertices, and let 𝑘 be the minimum number of prime factors of 𝜆(𝜎) among all coding sequences 𝜎 of 𝐺. Then 𝜃_𝑡(𝐺)=𝑘+𝑠.

Proof. Let 𝜎 be a coding sequence of 𝐺 such that the number of prime factors of 𝜆(𝜎) is 𝑘. Then by Lemma 6, 𝑘+𝑠=|𝑆_𝜎|⩾𝜃_𝑡(𝐺). Again let 𝑆 be a total clique covering of 𝐺 such that |𝑆|=𝜃_𝑡(𝐺). Then by Lemma 8, there is a coding sequence 𝜎 of 𝐺 such that 𝑆=𝑆_𝜎 and it follows from the proof of Lemma 8 that there are |𝑆|−𝑠 prime factors of 𝜆(𝜎). Thus, 𝑘⩽|𝑆|−𝑠=𝜃_𝑡(𝐺)−𝑠. Therefore, 𝜃_𝑡(𝐺)=𝑘+𝑠.

Definition 10. Let 𝐺=(𝑉,𝐸) be a graph with |𝑉|=𝑚 and 𝑠⩾0 isolated vertices. Let 𝑘=𝜃_𝑡(𝐺)−𝑠. Let 𝑆={𝐶₁,𝐶₂,…,𝐶_𝑘+𝑠} be a total clique covering of 𝐺 such that |𝑆|=𝜃_𝑡(𝐺) and |𝐶_𝑖|=1 for 𝑖=1,2,…,𝑠. Now as in the proof of Lemma 8, there are 𝑘! ways of assigning the first 𝑘 primes to the cliques 𝐶_𝑠+1,𝐶_𝑠+2,…,𝐶_𝑠+𝑘 to obtain at most 𝑘! different coding sequences. Let 𝜎[𝑆] be the least among them in the lexicographic ordering in ℝ^𝑚. Then 𝜎[𝑆] is called the coding sequence of 𝐺with respect to 𝑆. Let {𝑆₁,𝑆₂,…,𝑆_𝑟} be the set of all total clique coverings of 𝐺 such that |𝑆_𝑖|=𝜃_𝑡(𝐺) for all 𝑖=1,2,…,𝑟. Let 𝜎(𝐺) be the least element of {𝜎[𝑆_𝑖]|𝑖=1,2,…,𝑟} in the lexicographic ordering in ℝ^𝑚. Then 𝜎(𝐺) is called the code of the graph 𝐺.

For example, (2,3,3,5,5,6,6,6,7,11,385) and (2,3,3,5,5,7,10,10,10,11,231) are two coding sequences of the graph 𝐺 in Example 2 with respect to the total clique covering 𝑆={𝐶₁,𝐶₂,…,𝐶₅}. It is easy to see that 𝜃_𝑡(𝐺)=5 and 𝑆 is the only total clique covering with 5 cliques. The code of 𝐺 is (2,2,3,3,5,7,10,10,10,11,231). Note that for any graph 𝐺, we have 𝐺≅𝐺[𝜎(𝐺)].

Theorem 11. Let 𝐺₁ and 𝐺₂ be two graphs. Then 𝐺₁≅𝐺₂ if and only if 𝜎(𝐺₁)=𝜎(𝐺₂).

Proof. The proof follows from Definition 10.

Definition 12. Let 𝐺=(𝑉,𝐸) be a graph with 𝑠⩾0 isolated vertices. Let 𝑆={𝐶₁,𝐶₂,…,𝐶_𝑠+𝑘} be a total clique covering of 𝐺, where |𝐶_𝑖|=1 for 𝑖=1,2,…,𝑠. If 𝑣∉𝐶₁∪𝐶₂∪⋯∪𝐶_𝑠, let 𝑚(𝑣)=∏{𝑥_𝑗|𝑣∈𝐶_𝑠+𝑗} be a monomial in the polynomial semiring ℤ⁺₀[𝑥₁,𝑥₂,…,𝑥_𝑘] of 𝑘 indeterminates over the semiring of nonnegative integers with usual addition and multiplication. For any 𝑣∈𝐶_𝑖, 𝑖=1,2,…,𝑠, define 𝑚(𝑣)=1. Now we define 𝑓(𝐺,𝑆)=𝑓𝑥₁,𝑥₂,…,𝑥_𝑘=_𝑣∈𝑉𝑚(𝑣). (2) Then 𝑓(𝐺,𝑆) is said to be a polynomial representation of 𝐺 with respect to 𝑆.

Consider the graph 𝐺 in Example 2. Then 𝑓(𝐺,𝑆)=𝑥₁+2𝑥₂+2𝑥₃+𝑥₄+𝑥₅+3𝑥₁𝑥₂+𝑥₃𝑥₄𝑥₅ is a polynomial representation of 𝐺 with respect to 𝑆={𝐶₁,𝐶₂,𝐶₃,𝐶₄,𝐶₅}. Now let 𝐺=(𝑉,𝐸) be a graph, and let 𝜎=(𝜆₁,𝜆₂,…,𝜆_𝑚) be a coding sequence of 𝐺. From the construction of 𝑆_𝜎 (cf. Lemma 6, Definition 7) and by Definition 12, it follows that 𝑓(𝐺,𝑆_𝜎) can also be obtained from 𝜎 by replacing primes 𝑝_𝑖 by 𝑥_𝑖 (𝑖<𝑗⇔𝑝_𝑖<𝑝_𝑗) and commas by the addition symbol. It is important to note that the constant term of 𝑓(𝐺,𝑆) in Definition 12 is the number of isolated vertices of 𝐺.

Definition 13. Let 𝐺 be a graph and 𝜎(𝐺) the code of 𝐺. Then the polynomial representation of 𝐺 corresponding to 𝜎(𝐺), that is, 𝑓(𝐺,𝑆_𝜎(𝐺)), is called the normal polynomial representation or the canonical polynomial representation of 𝐺 and is denoted by 𝐹(𝐺).

The normal polynomial representation of the graph 𝐺 in Example 2 is given by 𝐹(𝐺)=2𝑥₁+2𝑥₂+𝑥₃+𝑥₄+𝑥₅+3𝑥₁𝑥₃+𝑥₂𝑥₄𝑥₅. (3) The following interesting observations are immediate from Definition 13.

Observation 1. A graph 𝐺 is disconnected if and only if 𝐹(𝐺)=𝑓₁+𝑓₂, where 𝑓₁ and 𝑓₂ are polynomials with no common variables between them. The same is true for 𝑓(𝐺,𝑆) for any total clique covering 𝑆 of 𝐺.

Observation 2. A graph 𝐺 is bipartite if and only if 𝐹(𝐺)=𝑓₁+𝑓₂, where monomials belonging to the same 𝑓_𝑖 have no common variables, for 𝑖=1,2. The same is true for 𝑓(𝐺,𝑆) for any total clique covering 𝑆 of 𝐺.

We now proceed to obtain a formula for 𝐹(𝐺(𝑛)).

Theorem 14. Let 𝑛=𝑝^𝑟1₁𝑝^𝑟2₂⋯𝑝^𝑟𝑘_𝑘∈𝑀, where the 𝑝_𝑖’s are distinct primes. Then 𝜃_𝑡(𝐺(𝑛))=𝑘 and there is only one total clique covering of 𝐺(𝑛) with precisely 𝑘 cliques.

Proof. If 𝑛 is prime, then the result is obvious. Suppose 𝑛 is not a prime number. Now the vertices of 𝐺(𝑛) correspond to all the divisors of 𝑛, greater than 1. So there are 𝑘 vertices labeled 𝑝_𝑖, 𝑖=1,2,…,𝑘. Clearly, no two of them can lie in the same clique as gcd(𝑝_𝑖,𝑝_𝑗)=1 for all 𝑖≠𝑗. So in any total clique covering of 𝐺(𝑛), these 𝑘 vertices will be in 𝑘 different cliques. In other words, any total clique covering has at least 𝑘 cliques. Now let 𝐶_𝑖={𝑣∈𝑉(𝐺(𝑛))|𝑝_𝑖divides𝑣}. Then 𝑆={𝐶_𝑖|𝑖=1,2,…,𝑘} is easily seen to be a total clique covering of 𝐺(𝑛). So we have a total clique covering consisting of precisely 𝑘 cliques. Hence, 𝜃_𝑡(𝐺(𝑛))=𝑘.
Now we show that there is only one total clique covering of 𝐺(𝑛) containing precisely 𝑘 cliques. Suppose there is another total clique covering 𝐷={𝐷₁,𝐷₂,…,𝐷_𝑘} of 𝐺 with exactly 𝑘 cliques. Here also, the vertices labeled 𝑝_𝑖, 𝑖=1,2,…,𝑘, are in distinct cliques. Without loss of generality, let 𝑝_𝑖∈𝐷_𝑖 for 𝑖=1,2,…,𝑘. Now for any 𝑖, 𝑝_𝑖 is not adjacent to those vertices which are not in 𝐶_𝑖. So 𝐷_𝑖⊆𝐶_𝑖 for all 𝑖=1,2,…,𝑘. Now suppose that for some 𝑗, there is a vertex 𝑣 in 𝐶_𝑗 which is not in 𝐷_𝑗. Now 𝐷 being a total clique covering, 𝑣 has to lie in at least one 𝐷_𝑙, where 𝑙≠𝑗. Since a total clique covering covers all the edges, the edge between 𝑣 and 𝑝_𝑗 will be covered. 𝑣∉𝐷_𝑗 implies that 𝑣 and 𝑝_𝑗 both lie in some 𝐷_𝑙, where 𝑙≠𝑗. However, this contradicts the fact that 𝑝_𝑗 is not adjacent to 𝑝_𝑙. So 𝐷_𝑖=𝐶_𝑖 for all 𝑖=1,2,…,𝑘 and hence 𝐷=𝑆.

Let 𝛼(𝐺) denote the maximum size of an independent set in a graph 𝐺. In general, 𝜃_𝑡(𝐺)⩾𝛼(𝐺) for any graph 𝐺. The proof of the following proposition is similar to that of Theorem 14 and so it is omitted.

Proposition 15. Let 𝐺 be a graph with 𝛼(𝐺)=𝑘. If {𝑣₁,𝑣₂,…,𝑣_𝑘} is an independent set and 𝑆={𝑆₁,𝑆₂,…,𝑆_𝑘} is a total clique covering of 𝐺 such that each 𝑣_𝑖 lies only in 𝑆_𝑖 among cliques in 𝑆 for each 𝑖=1,2,…,𝑘, then 𝜃_𝑡(𝐺)=𝛼(𝐺)=𝑘, and 𝑆 is the only total clique covering containing exactly 𝑘 cliques.

Now we provide a formula for 𝐹(𝐺(𝑛)).

Theorem 16. Let 𝑛=𝑝^𝑟1₁𝑝^𝑟2₂⋯𝑝^𝑟𝑘_𝑘∈𝑀, where the 𝑝_𝑖’s are distinct primes and 𝑟₁⩾𝑟₂⩾⋯⩾𝑟_𝑘⩾1, 𝑘⩾1. Then 𝐹(𝐺(𝑛)) contains all the monomials 𝑥_𝑖1𝑥_𝑖2⋯𝑥_𝑖𝑠, where {𝑖₁,𝑖₂,…,𝑖_𝑠}⊆{1,2,…,𝑘}, 𝑠⩾1 with the coefficient 𝑟_𝑖1𝑟_𝑖2⋯𝑟_𝑖𝑠, that is, 𝐹(𝐺(𝑛))=_{𝑖1<𝑖2<⋯<𝑖𝑠𝑖∅≠1,𝑖2,…,𝑖𝑠⊆{}1,2,…,𝑘}𝑟_𝑖1𝑟_𝑖2⋯𝑟_𝑖𝑠𝑥_𝑖1𝑥_𝑖2⋯𝑥_𝑖𝑠, (4) unless 𝑛 is prime. If 𝑛 is prime, then 𝐹(𝐺(𝑛))=1.

Proof. If 𝑛 is prime, then 𝐺(𝑛)≅𝐾₁ and so 𝐹(𝐺(𝑛))=1. Suppose 𝑛 is not prime. Now by Theorem 14, 𝜃_𝑡(𝐺(𝑛))=𝑘 and there is only one total clique covering, say, 𝑆 of 𝐺(𝑛) with precisely 𝑘 cliques. Let 𝑆={𝐶₁,𝐶₂,…,𝐶_𝑘}, where 𝐶_𝑖={𝑣∈𝑉(𝐺(𝑛))|𝑝_𝑖divides𝑣} for 𝑖=1,2,…,𝑘. So 𝜎(𝐺(𝑛)) is 𝜎[𝑆]. First, let 𝑘>1. Now 𝜎[𝑆] is the least (in the lexicographic ordering) among the coding sequences of 𝐺 obtained from 𝑆. By Definition 10, any such coding sequence involves the primes {𝑡₁,𝑡₂,…,𝑡_𝑘}, where 𝑡_𝑖 is the 𝑖th prime. Now for any 𝑝_𝑖, there are 𝑟_𝑖 elements (namely, 𝑝_𝑖,𝑝²_𝑖,…,𝑝^𝑟𝑖_𝑖) which belong to only the clique 𝐶_𝑖. So considering the labellings and the fact that 𝑟₁⩾𝑟₂⩾⋯⩾𝑟_𝑘⩾1, it is easy to see that the coding sequence will be the least in the lexicographic ordering if we assign 𝑡_𝑖 to the clique C_𝑖. So the vertex corresponding to a number 𝑝^𝑞𝑖1_𝑖1𝑝^𝑞𝑖2_𝑖2⋯𝑝^𝑞𝑖𝑠_𝑖𝑠 (1⩽𝑞_𝑖𝑗⩽𝑟_𝑖𝑗 for 𝑗=1,2,…,𝑠 where 1⩽𝑠⩽𝑘) is assigned the label 𝑡_𝑖1𝑡_𝑖2⋯𝑡_𝑖𝑠. In other words, to find out 𝜎(𝐺(𝑛)), we first consider the set of all divisors {𝑚₁,𝑚₂,…,𝑚_𝑟} (say) of 𝑛, greater than 1, where 𝑟=(𝑟₁+1)(𝑟₂+1)⋯(𝑟_𝑘+1)−1. Then in each 𝑚_𝑗, we first replace 𝑝_𝑖 by 𝑡_𝑖, then make the resultant entries square-free and arrange them in nondecreasing order. This gives us 𝜎(𝐺(𝑛))=(𝜆₁,𝜆₂,…,𝜆_𝑟), where each 𝜆_𝑖 is a product of primes of the form 𝑡_𝑖1𝑡_𝑖2⋯𝑡_𝑖𝑠, (1⩽𝑠⩽𝑘) and this particular number repeats, say, 𝜈 times in the sequence 𝜎(𝐺(𝑛)), where 𝜈 is the number of divisors of 𝑛 (greater than 1) which are of the form 𝑝^𝑞𝑖1_𝑖1𝑝^𝑞𝑖2_𝑖2⋯𝑝^𝑞𝑖𝑠_𝑖𝑠, 1⩽𝑞_𝑖𝑗⩽𝑟_𝑖𝑗 for 𝑗=1,2,…,𝑠; that is, 𝜈=∏^𝑠_𝑗=1𝑟_𝑖𝑗.
Thus, by Definition 13 we have 𝐹(𝐺(𝑛))=∑_{𝑖1<𝑖2<⋯<𝑖𝑠,∅≠{𝑖1,𝑖2,…,𝑖𝑠}⊆{1,2,…,𝑘}}𝑟_𝑖1𝑟_𝑖2⋯𝑟_𝑖𝑠𝑥_𝑖1𝑥_𝑖2⋯𝑥_𝑖𝑠.
Finally, if 𝑘=1,𝑟₁>1, then 𝐺(𝑛)≅𝐾_𝑟1. So 𝐹(𝐺(𝑛))=𝑟₁𝑥₁, which satisfies the aforementioned formula. This completes the proof.

For example, 𝐹(𝐺(𝑝²𝑞𝑟))=𝐹(𝐺(60))=2𝑥₁+𝑥₂+𝑥₃+2𝑥₁𝑥₂+2𝑥₁𝑥₃+𝑥₂𝑥₃+2𝑥₁𝑥₂𝑥₃, where 𝑝, 𝑞, and 𝑟 are distinct primes. Further, one may easily verify codes and normal polynomial representations for the following special classes of graphs:(i)𝜎(𝐾₁)=(1), 𝐹(𝐾₁)=1, 𝜎(𝐾_𝑛)=(2,2,…,2) (𝑛 times), 𝐹(𝐾_𝑛)=𝑛𝑥₁ for 𝑛⩾2, (ii)𝜎(𝑃_𝑛)=(𝑝₁,𝑝₂,𝑝₁𝑝₃,𝑝₂𝑝₄,…,𝑝_𝑛−3𝑝_𝑛−1,𝑝_𝑛−2𝑝_𝑛−1), 𝐹(𝑃_𝑛)=𝑥₁+𝑥₂+∑^𝑛−3_𝑖=1𝑥_𝑖𝑥_𝑖+2+𝑥_𝑛−2𝑥_𝑛−1 for 𝑛⩾3, (iii)𝜎(𝐶_𝑛)=(𝑝₁𝑝₂,𝑝₁𝑝₃,𝑝₂𝑝₄,…,𝑝_𝑛−2𝑝_𝑛,𝑝_𝑛−1𝑝_𝑛), 𝐹(𝐶_𝑛)=𝑥₁𝑥₂+𝑥₁𝑥₃+∑^𝑛−2_𝑖=1𝑥_𝑖𝑥_𝑖+2+𝑥_𝑛−1𝑥_𝑛 for 𝑛⩾4, where 𝑝_𝑖 is the 𝑖th prime for 𝑖=1,2,…,𝑛 and 𝐾_𝑛, 𝑃_𝑛, and 𝐶_𝑛 are, respectively, the complete graph, the path, and the cycle with 𝑛 vertices.

3. Conclusion

There are some representations of simple undirected graphs in terms of adjacency matrices, adjacency lists, and unordered pairs which are not unique for isomorphic graphs. There are some other instances for unique representations [2, 3]. In the nauty algorithm, McKay [2] defined a canonical isomorph [4] which is a graph rather than a sequence of integers as the code of a graph introduced here. The importance of the code 𝜎(𝐺) is its uniqueness and its simple form. It is the same for any set of isomorphic graphs. The determination of 𝜎(𝐺) is not always easy, but once it is obtained for a graph, it becomes the characteristic of the graph. The authors believe that further study of the code and the normal polynomial representation of a simple undirected graph will be helpful in further research on graph theory. The purpose of this paper is to communicate these interesting observations to all graph theorists.

Acknowledgment

The authors gratefully acknowledge the learned referees for their kind suggestions and comments which enriched the paper.