Unitary Cayley graphs of Dedekind domain quotients

Peer review under responsibility of Kalasalingam University.

Abstract

If is a commutative ring with unity, then the unitary Cayley graph of , denoted , is defined to be the graph whose vertex set is and whose edge set is . When is a Dedekind domain and is an ideal of such that is finite and nontrivial, we refer to as a generalized totient graph. We study generalized totient graphs as generalizations of the graphs , which have appeared recently in the literature, sometimes under the name Euler totient Cayley graphs. We begin by generalizing to Dedekind domains the arithmetic functions known as Schemmel totient functions, and we use one of these generalizations to provide a simple formula, for any positive integer , for the number of cliques of order in a generalized totient graph. We then proceed to determine many properties of generalized totient graphs such as their clique numbers, chromatic numbers, chromatic indices, clique domination numbers, and (in many, but not all cases) girths. We also determine the diameter of each component of a generalized totient graph. We correct one erroneous claim about the clique domination numbers of Euler totient Cayley graphs that has appeared in the literature and provide a counterexample to a second claim about the strong domination numbers of these graphs.

Keywords:

• Unitary Cayley graph
• Dedekind domain
• Schemmel totient function
• Clique
• Domination

1 Introduction

Throughout this paper, we will let be an arbitrary Dedekind domain. For nonzero ideals and of , we will make repeated use of the fact that regardless of whether or not and are relatively prime ideals. Furthermore, if factors into powers of prime ideals as , then if and only if for each . When is finite, we will refer to as the index of in . If and is finite and nontrivial, then we will let denote the minimum of as ranges over all prime ideal divisors of .

The well-known Euler totient function maps a positive integer to the number of positive integers that are less than or equal to and relatively prime to . In other words, . In 1869, V. Schemmel introduced a class of functions , now known as Schemmel totient functions, which generalize the Euler totient function. For all positive integers and , counts the number of positive integers such that for all . Thus, . We will convene to let for all . For each , is a multiplicative arithmetic function that satisfies (1) for all primes and positive integers [1]. We may consider an extension of the Euler totient function to Dedekind domains, defining whenever is an ideal of such that is finite (we will use the symbol to represent the traditional Euler totient function whose domain is , and we will use to represent this mapping from quotients of Dedekind domains to ). Thus, for all . Later, we will define two classes of functions that will each serve to extend the Schemmel totient functions to Dedekind domains.

Our primary goal is to study properties of certain unitary Cayley graphs. If is a commutative ring with unity, then the unitary Cayley graph of , denoted , is defined to be the graph whose vertex set is and whose edge set is . The unitary Cayley graphs for have been named “Euler totient Cayley graphs” [2^[3]–4]. Several researchers have shown that the graph contains exactly triangles [2,5,6], and Manjuri and Maheswari have studied Euler totient Cayley graphs in the context of domination parameters [3,4]. Klotz and Sander have studied, among other properties, the diameters and eigenvalues of Euler totient Cayley graphs [6], and their paper gives a list of references to other results related to these graphs.

We define a generalized totient graph to be a unitary Cayley graph , where is an ideal of the Dedekind domain and is finite and nontrivial. We seek to gain information about many of the properties of generalized totient graphs. In particular, we will use one of our two extensions of the Schemmel totient functions to give a formula, for each positive integer , for the number of cliques of order in a given generalized totient graph. This formula, which apparently has not yet appeared anywhere in the literature even for Euler totient Cayley graphs, will allow us to determine the clique domination numbers of generalized totient graphs and correct an erroneous claim that Manjuri and Maheswari have made regarding this topic. We will build upon the work of Klotz and Sander, who have determined the diameters of Euler totient Cayley graphs. We end the paper with suggestions for further research and a counterexample to a claim that Manjuri and Maheswari have made regarding the strong domination numbers of Euler totient Cayley graphs.

2 Extending the schemmel totient functions

Our first extension of the Schemmel totient functions is inspired by our original definition of , for any given , as the number of positive integers such that for all . Implicit in the following definition is the fact that every Dedekind domain has a unity element, which we will denote 1. Furthermore, a positive integer , when used to denote an element of a Dedekind domain, will be understood to represent the sum .

Definition 2.1

Let be an ideal of such that is finite. For any positive integer , we define the set by Furthermore, we define by .

Remark 2.1

Setting and in Definition 2.1 yields . Observe that . Also, note that , so .

The following two theorems show that the functions can be evaluated using a formula similar to (1(1) ).

Theorem 2.1

Let and be relatively prime nonzero ideals of such that and are finite. Then for all positive integers .

Proof

Fix some positive integer . Consider the natural ring homomorphisms and defined by and . Because and are relatively prime, we know that the function defined by is a ring isomorphism. Now, if and only if for all . This occurs if and only if and , which occurs if and only if . Thus, there is a bijection between and , so . □

Theorem 2.2

Let be a prime ideal of such that is finite. Let and be positive integers. Then

Proof

If we define a ring homomorphism by , then we see that if and only if . More generally, for all if and only if for all . Therefore, if and only if . As is clearly surjective and , we know that is a -to-1 mapping, where . This shows that . Since is a prime ideal of the Dedekind domain , is maximal. Consequently, is a field. It follows that if and if . Thus,  □

Let , where are distinct prime ideals of and , are positive integers. Then we may use Theorem 2.1 repeatedly to write for any positive integer . Theorem 2.2 then allows us to evaluate for each positive integer . Note that is defined if is finite. In this case, must be a field (any finite integral domain is a field), so an argument similar to that used in the proof of Theorem 2.2 shows that

We have established one extension of the Schemmel totient functions that is interesting in its own right, but we will see that the following slightly different extension will prove itself much more useful for our purposes later.

Definition 2.2

Let be a nonnegative integer. For each nonzero ideal of finite index in , we may define the nonnegative integer by the following rules:

(a)

.

(b)	If is a prime ideal of finite index in and is a positive integer, then

(c)	If and are relatively prime nonzero ideals of finite index in , then .

Remark 2.2

First, note that we can evaluate for any nonzero ideal of finite index in by simply decomposing into a product of powers of prime ideals and combining the given rules. Then if and only if . It is easy to see that and for any nonzero ideal of finite index in . Finally, note that if we set , then for any positive integers and .

3 Enumerating cliques in generalized totient graphs

In this section, we will prove our central result, which provides a formula for the number of cliques of order in any generalized totient graph. From now on, we will always let denote a nonzero ideal of finite index in the Dedekind domain .

We will need the following lemma.

Lemma 3.1

Let be a nonzero ideal of such that is finite and nontrivial, and let be a positive integer. Then if and only if there exist elements of such that for all distinct . Furthermore, if , then, for any such choice of , there are exactly elements of that satisfy for all .

Proof

First, by way of contradiction, suppose that and that are elements of such that for all distinct . As , there must be some prime ideal divisor of such that . By the pigeonhole principle, we must have for some distinct . However, this is a contradiction because it implies that , which then implies that .

Now, suppose . Let us write , where , are distinct prime ideals of and are positive integers. For each positive integer , we may write and . For each positive integer , the Chinese remainder theorem guarantees that it is possible to find some such that for all positive integers . Then, for all distinct and all , we have , so . This means that for all distinct .

Finally, suppose that and that satisfy for all distinct . For each , let be the set of elements of such that for all (observe that the choices of and as representatives of their cosets do not affect whether or not because ). By the Chinese remainder theorem, any element of is uniquely determined by the cosets of the ideals that contain the representative . Therefore, the number of ways to choose an element of that satisfies for all is equal to . Fix , and consider the natural homomorphism given by . We know that is a -to-1 mapping, where . An element of is in if and only if for all . In other words, is the preimage of the set under . Hence, Using Definition 2.2, we see that the number of elements of that satisfy for all is  □

Theorem 3.1

For any positive integer , the number of cliques of order in the graph is given by the expression

Proof

Let be the number of cliques of of order . The result is trivial if because , which is the number of vertices of . Now, suppose that . We will show that ; the theorem will then follow by induction on . If , then of course because there are no cliques of order . Therefore, let us assume that there is at least one clique of order in , say . Let the vertices in be . Then for all distinct because is a clique. By Lemma 3.1, there are exactly elements of that satisfy for all . In other words, there are exactly vertices of that we may annex to to make a clique of order . This counts each clique of order a total of times because there are subcliques of order in every clique of order . Hence, as desired. □

Corollary 3.1

Let and be positive integers with . The number of cliques of order in the Euler totient Cayley graph is .

Observe that if we set in Corollary 3.1, we recover the previously-discovered formula for the number of triangles in . Our proof of this formula seems much more natural and illuminating than those already in existence [2,5,6]. Before proceeding to uncover some additional properties of generalized totient graphs, we pause to note an interesting divisibility relationship that arises as a corollary of Theorem 3.1.

Corollary 3.2

For any positive integer , we have

4 Other properties of generalized totient graphs

For any graph , let and denote the chromatic number and the chromatic index of , respectively. Let , , , and denote, respectively, the girth, diameter, maximum degree, and clique number of . A set of vertices of is said to dominate if every vertex in is either in or is adjacent to at least one element of . The clique domination number is the smallest positive integer such that there exists a clique of of order that dominates (provided some dominating clique exists).

We will continue to let denote a nonzero ideal of the Dedekind domain such that is both finite and nontrivial. We begin with a fairly basic lemma concerning unitary Cayley graphs of commutative rings with unity. A symmetric graph is a graph such that if are vertices of with adjacent to and adjacent to , then there exists an automorphism of that maps to and maps to .

Lemma 4.1

Let be a commutative ring with unity. The unitary Cayley graph is symmetric and -regular, where .

Proof

Choose some such that is adjacent to and is adjacent to . Define a function by Observe that and . It is straightforward to see that is a bijection because and are units in . Now, let and be any adjacent vertices in . It follows from the fact that , , and are all units in that . Hence, and are adjacent. Similarly, maps nonadjacent vertices to nonadjacent vertices. This shows that is an automorphism, so is symmetric. As any symmetric graph is regular and the degree of the vertex 0 is , we see that is -regular. □

One of the first results proved about unitary Cayley graphs states that if , then is bipartite if and only if is even [5]. The proof is fairly straightforward, and it generalizes immediately to generalized totient graphs. For this reason, we record the following fact and omit the proof.

Fact 4.1

The generalized totient graph is bipartite if and only if .

Another standard result concerning unitary Cayley graphs states that the clique number and the chromatic number of are both equal to the smallest prime factor of [6]. Again, the proof generalized in a straightforward manner, so we omit the proof of the next fact. We will remark, however, that the fact that follows immediately from Theorem 3.1 and the fact that if and only if .

Fact 4.2

We have .

Before attempting to determine the chromatic indices of generalized totient graphs, we need two lemmas that should make the proof of the following theorem relatively painless.

Lemma 4.2

Let be a positive even integer. Let be a simple -partite graph with partite sets , all of the same cardinality, such that for all distinct and any , is adjacent to exactly as many vertices in as it is to vertices in . Then .

Proof

Because of the trivial inequality , it suffices to exhibit a proper edge-coloring of with colors. Let be a vertex of of degree . Without loss of generality, we may assume that . Suppose is adjacent to exactly vertices in . Then, by property , is adjacent to exactly vertices in for each . As is not adjacent to any vertices in , we see that . Hence, we may label our colors , where ranges over the set and ranges over the set . Now, let be the vertices of the complete graph . It is well-known that, because is even, it is possible to properly color the edges of with colors, say . For any distinct , let be the integer such that is the color used to color the edge connecting and .

We now describe how to color the edges of . For any distinct , let be the subgraph of induced by the vertices in . Every such graph is clearly a bipartite graph with maximum degree at most . Hence, König’s line coloring theorem implies that, for any distinct , it is possible to properly color the edges of with only the colors in the set . Doing so for all subgraphs yields a proper coloring of with colors. □

Lemma 4.3

Let be a positive integer. Any -regular simple graph with an odd number of vertices has chromatic index .

Proof

Let be a -regular simple graph with vertices, where is an odd positive integer. If there exists a proper edge-coloring of with colors, then no color can be used more than times. This implies that the number of edges of cannot exceed . As the number of edges of is , we have . Consequently, . By the definition of , there exists a proper edge-coloring of with colors, so . Vizing’s theorem states that , so we conclude that . □

Theorem 4.1

If has a prime ideal divisor whose index in is a power of 2, then . Otherwise, .

Proof

First, suppose has a prime ideal divisor such that for some positive integer . Let . Let us write , where is a positive integer and . Define a homomorphism by , and, for each positive integer , let . The sets , all of which have the same cardinality, partition the vertices of . Furthermore, no two vertices in the same set are adjacent. Hence, satisfies property of Lemma 4.2. Now, fix some distinct and some vertex . There are exactly elements of such that , and every one of those elements satisfies because and . Also, if we view the set as a coset of the subgroup of , then we see that there are exactly elements of such that . A vertex of is an element of that is adjacent to if and only if , , and . Because , we see that there are exactly such vertices. This number does not depend on the choice of (so long as ), so satisfies property of Lemma 4.2. Therefore, by Lemmas 4.1 and 4.2, .

Now, suppose that does not have a prime ideal divisor whose index in is a power of 2. Let us write , where are prime ideals of and are positive integers. For each positive integer , is a finite field, so must be a power of an odd prime. Hence, is odd, so has an odd number of vertices. Using Lemma 4.1 and Lemma 4.3, we conclude that . □

Theorem 4.2

Let denote the number of distinct prime ideal divisors of . If is a prime ideal, then . If is not a prime ideal and , then . If is not prime and , then does not exist. Furthermore, if exists, then every clique of of order dominates .

Proof

Because is a Dedekind domain, is a field if and only if is prime. In other words, is complete if and only if is prime. Therefore, if is prime, any single vertex of forms a dominating clique. Now, suppose is not prime. Let be the prime ideal divisors of , and assume that is a clique of of order . If , then we know we may find some vertex of other than that is not adjacent to because is regular and not complete. Therefore, no clique of order 1 dominates . Suppose . By the Chinese remainder theorem, we know that we may find some such that for all positive integers . This implies that is not adjacent to any element of . Then, because any vertex in is adjacent to all other vertices in , cannot be in . Thus, does not dominate , so we conclude that no clique of order can dominate when is not prime.

Now, suppose is not prime and . Again, let be the prime ideal divisors of . There exists at least one clique of of order because (by Fact 4.2), so we may let be an arbitrary clique of of order . We will show that dominates . Suppose, for the sake of finding a contradiction, that there is some vertex that is not adjacent to any of the vertices of . By the pigeonhole principle, there must be some prime ideal divisor of and some distinct such that and . Then , which contradicts the fact that because is a clique.

Finally, suppose is not prime and . Because the clique number of is by Fact 4.2, we see that there are no cliques of order larger than . Thus, no clique of can dominate , so does not exist. □

If , then because by Fact 4.2. On the other hand, if , then Fact 4.1 implies that contains no odd cycles. Therefore, if , then . The following theorem shows that for many generalized totient graphs .

Theorem 4.3

Let be a prime principal ideal of , and let be an ideal of that is not contained in . Let , where is a positive integer. If or if , then has a cycle of length 4.

Proof

Suppose or . Let be an element of that is not in . Consider the vertices , , , and . Suppose and are not adjacent. Then is an element of some prime ideal divisor of . Since , . Hence, is a prime ideal divisor of . This implies that because . We then have , so is a prime ideal divisor of . As is a prime ideal, we must have , which contradicts our hypothesis that . Hence, and are adjacent. Similar arguments show that is adjacent to , is adjacent to , and is adjacent to . Therefore, if , , , and are all distinct, they form a cycle of length 4. We know that , , , and because no vertex of can be adjacent to itself. Hence, it suffices to show that and .

Assume . As and , . This implies that , so . Thus, . Similarly, , so .

Suppose, now, that . Then , so . This shows that . Similarly, , so . This implies that . □

Corollary 4.1

If is an integer, then

Proof

If , then it follows from the paragraph immediately preceding Theorem 4.3 that because . It is easy to see that because is a cycle of length 6. Assume, now, that and . Because , . Write for some positive integers and with odd. Setting and in Theorem 4.3 shows that there is a cycle of length 4 in , so . □

5 Diameters and disconnectedness

Klotz and Sander have determined the diameters of all Euler totient Cayley graphs [6], so we will do the same for generalized totient graphs. We wish to acknowledge that the proofs of Lemma 5.1, Theorem 5.1, and Theorem 5.3 are inspired by proofs that Klotz and Sander used to establish similar results in the specific case when .

Definition 5.1

For each prime ideal of and element of , define by Let be the (unique) factorization of into a product of powers of distinct prime ideals. We define by

The function clearly depends on the choice of and the choice of , but we trust that this will not lead to confusion.

Lemma 5.1

For any , the vertices and of have common neighbors.

Proof

As before, we will let be the factorization of into a product of powers of distinct prime ideals. Fix some , and note that a vertex is a common neighbor of and in if and only if and for all . For each , let be the number of common neighbors of and in . It follows from the Chinese remainder theorem that the number of common neighbors of and in is . Now, choose some so that we may evaluate . We use the natural homomorphism defined by , which we know is a -to-1 mapping with . A vertex is a common neighbor of and in if and only if and . Therefore, if we wish to choose to be a common neighbor of and in , then there are exactly elements of that cannot be the image of under . We see that . Hence,  □

Theorem 5.1

If is a prime ideal of , then . If is not a prime ideal of and has no prime ideal divisors of index 2, then .

Proof

If is a prime ideal of , then is a field. It follows that is complete, so .

Suppose is not a prime ideal of and has no prime ideal divisors of index 2. Let be a prime ideal divisor of . We may choose some with . The vertices and are distinct and nonadjacent. This implies that . Now, for any , it is easy to see that because the index of each prime ideal divisor of is at least 3. It follows that any vertices and have a common neighbor, so . □

Theorem 5.2

Suppose that has exactly distinct prime ideal divisors of index 2 in , where . Then has exactly disconnected components. All such components are bipartite graphs that are isomorphic to each other and whose diameters are at most 3.

Proof

For any vector , let denote the th coordinate of . Let be the prime ideal divisors of of index 2. We define a function by specifying that for each and , we have It is easy to verify that is well-defined. If we write , then the collection of sets forms a partition of into subsets. Let denote the vector in whose coordinates are all 1’s. It is easy to see that for all . Furthermore, . Therefore, two vertices and of can only be adjacent if . In other words, for each , the vertices in are only adjacent to vertices in . Let be a set of vectors in with the property that if and only if , and write . For each , write , and let be the subgraph of induced by . Because no vertex in is adjacent to any vertex in when , we see that is the union of the disconnected subgraphs . Furthermore, each subgraph is bipartite because we may partition the set of vertices of into the sets and .

Choose some and some . Observe that . In other words for all . This implies that , so Lemma 5.1 tells us that and have a common neighbor. The same argument shows that any two vertices in must have a common neighbor. Therefore, the subgraph is connected and has diameter at most 3. As was arbitrary, this shows that each subgraph is connected and has diameter at most 3. Finally, the subgraphs are isomorphic to each other because is symmetric by Lemma 4.1. □

Theorem 5.1 gives the diameter of when is not divisible by a prime ideal of index 2. When is divisible by a prime ideal of index 2, Theorem 5.2 tells us that could be a union of several disconnected components. The following theorem determines the diameter of each component of when is divisible by a prime ideal of index 2.

Theorem 5.3

Let . Suppose , where are distinct prime ideals of index 2 in , are positive integers, and is an ideal of that is not divisible by a prime ideal of index 2. If and for all , then the diameter of each component of is 1. If and for some , then the diameter of each component of is 2. If , then the diameter of each component of is 3.

Proof

Preserve the notation from the proof of Theorem 5.2, and let . First, suppose that . The number of vertices in is . We know from the proof of Theorem 5.2 that and form two partite sets of the bipartite component . If and , then . This implies that for all , so and are adjacent. It follows that is a complete bipartite graph. Theorem 5.2 tells us that has disconnected components that are all isomorphic to each other, so must contain exactly vertices. If for all , then has only two vertices. In this case, the diameter of is 1 because is isomorphic to the complete bipartite graph . If for some , then . In this case, is a complete bipartite graph with at least four vertices, so it must have diameter 2. Because all of the components of are isomorphic to , this completes the proof of the case in which .

Suppose, now, that . The Chinese remainder theorem guarantees that we may choose some such that for all . There exists some such that . Because and , we know that and are in the same component . The vertices and are not adjacent because . Also, Lemma 5.1 implies that and have no common neighbors because . Hence, the diameter of is at least 3. By Theorem 5.2, the diameter of must be equal to 3. The proof then follows from the fact that all components of are isomorphic to . □

6 Strong colorings of generalized totient graphs

Suppose we are given a positive integer and a graph with vertices. Let be the least nonnegative integer such that , and let be the graph that results from adding isolated vertices to . We say that is strongly -colorable if, for any given partition of the vertices of into subsets of size , it is possible to properly color the vertices of so that each color appears exactly once in each subset of the partition. Observe that if is simple and has some vertex of degree at least , then we can choose to partition the vertices of into subsets of size so that one of the subsets is contained in the neighborhood of . As no vertex in the neighborhood of can have the same color as in a proper coloring of , we see that any simple graph with a vertex of degree at least cannot be strongly -colorable. The strong chromatic number of a graph , denoted , is the smallest positive integer such that is strongly -colorable. It follows from the preceding discussion that must be greater than if is simple.

We say that an edge-coloring of a graph is strong if any two distinct edges with adjacent endpoints are colored differently. The strong chromatic index of a graph , denoted , is the smallest positive integer such that it is possible to strongly edge-color with colors.

In this section, we briefly study the strong chromatic numbers and strong chromatic indices of some generalized totient graphs.

Theorem 6.1

Let be a nonzero prime ideal of such that is finite, and let be a positive integer. Then .

Proof

For convenience, we write . Note that has vertices. We know that , and Theorem 2.2 tells us that . Let for some positive integer , and assume by way of contradiction that is strongly -colorable. It is easy to see that is the least nonnegative integer such that is divisible by . Let be the graph that results from adding isolated vertices to . Let , and let , where . Then because . If we let be the set of vertices of that are not in , then we see that the two sets and form a partition of the set of vertices of into subsets of size . Because is strongly -colorable, it is possible to properly color the vertices of so that each color appears exactly once in and exactly once in .

Let be colored with the color . We know that must be used to color exactly one vertex . Suppose . Then because . However, , so . This shows that , so is adjacent to . This is a contradiction because we assumed the coloring to be proper. Thus, each of the colors used to color the elements of must be used to color one of the elements of the set . However, this is impossible because . Hence, we must have . Clearly, is strongly -colorable, so . □

Theorem 6.2

Let , , and be nonzero ideals of such that and are prime, , and are finite and nontrivial, and . Let be a positive integer. We haveand

Proof

By Theorem 4.2, any two adjacent vertices of dominate . Therefore, in any strong edge-coloring of , no two distinct edges can have the same color. This means that any strong edge-coloring of must use exactly colors because has edges.

Note that since . Consider the graph , which has an edge set of size Choose some with . For any vertex of , let . For any edge of , let . Notice that, for any vertex of , we have because . Furthermore, it is easy to see that is not equal to or adjacent to . This shows that for all edges of . Let us color the edges of so that two distinct edges and have the same color if and only if . Because each color is used to color two of the edges of , this coloring uses colors.

To show that this coloring is strong, suppose and are adjacent vertices of . We stated already that is not adjacent to . Because and , we must have . Therefore, , which means that . Hence, is not adjacent to . By the same token, is not adjacent to or . This shows that, for any edge , the only other edge with the same color as cannot have an endpoint adjacent to an endpoint of . Thus, the coloring is strong. □

7 Erratum

We use this section to briefly expose two mistakes that have appeared in the literature. First, Manjuri and Maheswari have made the claim that if is a composite odd integer, then exists if and only if has 2 or fewer distinct prime factors [3]. However, setting and in Theorem 4.2 shows that their claim is false.

We now provide a counterexample to Theorem 4.3 of a different paper of Manjuri and Maheswari [4]. We hope to encourage the reader to discover a correct solution to this interesting problem and perhaps even generalize the solution in order to solve an analogous problem concerning generalized totient graphs. We have been able to obtain partial results in this direction, but have been unable to solve the problem in general.

The domination number of a graph is the size of the smallest dominating set of . The paper states that if a composite integer is not twice a prime, then is the smallest positive integer such that every set of consecutive integers contains an element that is not relatively prime to (this number is often denoted , and is called Jacobsthal’s function). One may verify that the set dominates (we let ). Therefore, (in fact, it can be shown that for any distinct primes ). However, , so the claim is false.

8 Concluding remarks

We wish to acknowledge the potential to generalize and strengthen the preceding results. For example, one might wish to study the infinite graphs that arise from eliminating the restriction that be finite. Alternatively, one might attempt to gather information about analogues of gcd-graphs [6] in Dedekind domains. There is certainly room for strengthening Theorems 6.1 and 6.2, which only apply to generalized totient graphs with specific properties. Theorem 4.3 leads us to inquire about which generalized totient graphs have girths not equal to 3 or 4. Finally, we note that we have obviously not exhausted all of the graph parameters of generalized totient graphs that one might wish to study.

Notes

Peer review under responsibility of Kalasalingam University.