Vertex-Relaxed Graceful Labelings of Graphs and Congruences Florin Aftene Western Kentucky University, [email protected]

Vertex-Relaxed Graceful Labelings of Graphs and Congruences Florin Aftene Western Kentucky University, Darwoodfez@Gmail.Com

Western Kentucky University TopSCHOLAR®

Masters Theses & Specialist Projects Graduate School

Spring 2018 Vertex-Relaxed Graceful Labelings of Graphs and Congruences Florin Aftene Western Kentucky University, [email protected]

Follow this and additional works at: https://digitalcommons.wku.edu/theses Part of the Discrete Mathematics and Combinatorics Commons

Recommended Citation Aftene, Florin, "Vertex-Relaxed Graceful Labelings of Graphs and Congruences" (2018). Masters Theses & Specialist Projects. Paper 2664. https://digitalcommons.wku.edu/theses/2664

This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. VERTEX-RELAXED GRACEFUL LABELINGS OF GRAPHS AND CONGRUENCES

A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky

In Partial Fulﬁllment Of the Requirements for the Degree Master of Science

By Florin G. Aftene

May 2018

ACKNOWLEDGMENTS

I would like to thank Dr. Dominic Lanphier for his help and guidance with the masters thesis. I could have not done it without his help. I would like to thank my wife Jennifer Aftene for believing in me and supporting me throughout my masters program. Finally, I would like to thank my mother Florentina Hiller Shropshire for encouraging me to pursue my goal of getting a masters degree and helping me to get over the fear of the unknown. With out the guidance and encouragement of Dr. Lanphier, Jennifer Aftene, and Florentina Hiller Shropshire I would have not been able to get into the masters program let alone complete it. Once again thank you.

iii Contents

1 Introduction 1

2 Deﬁnitions and Conjectures 3 2.1 Deﬁnitions ...... 3

3 Arithmetic Properties of Vertex-Relaxed Graceful Labelings 7

4 Faulhaber’s Formula and a Generalization 16 4.1 Prerequisites for Faulhaber’s Formula ...... 16 4.2 Introduction to Faulhaber’s Formula ...... 18 4.3 Proof of Faulhaber’s Formula ...... 18 4.4 Introduction to an Extension of Faulhaber’s Formula ...... 21 4.5 A Generalization of Faulhaber’s Formula ...... 24

5 A Further Criterion for Vertex-Relaxed Graceful Labelings 29 5.1 An Application of Proposition 5.0.2 ...... 34

6 Conclusion 38

iv List of Figures

2.1 Cycle graphs (C3 − C6)...... 4

2.2 Kn Graphs [18] ...... 4 2.3 Example of a caterpillar graph ...... 5 2.4 Three examples of graceful graphs and their graceful labelings. . . . .6 2.5 Two examples of non-graceful graphs...... 6

5.1 Graceful Labeling Example 1[15] ...... 34 5.2 Graceful Labeling Example 2[15] ...... 35 5.3 Graceful Labeling Example 2[15] ...... 36 5.4 Graceful Labeling Example 2[15] ...... 37

v VERTEX-RELAXED GRACEFUL LABELINGS OF GRAPHS AND CONGRUENCES Florin G. Aftene May 2018 40 Pages Directed by: Dr. Dominic Lanphier, Dr. Attila Por, and Dr. Claus Ernst Department of Mathematics Western Kentucky University

A labeling of a graph is an assignment of a natural number to each vertex of a graph. Graceful labelings are very important types of labelings. The study of graceful labelings is very diﬃcult and little has been shown about such labelings. Vertex-relaxed graceful labelings of graphs are a class of labelings that include graceful labelings, and their study gives an approach to the study of graceful labelings. In this thesis we generalize the congruence approach of Rosa to obtain new criteria for vertex-relaxed graceful labelings of graphs. To do this, we generalize Faulhaber’s Formula, which is a famous result about sums of powers of integers.

vi Chapter 1

Introduction

Not all graphs have a graceful labeling. In fact, as mentioned in [16], most graphs do not have any graceful labeling. If there does exist a graceful labeling of a given graph, then that graph is said to be graceful. A tree is a connected graph without any cycles. A long standing and important conjecture in graph theory is called the Graceful Tree Conjecture (GTC) that originated in the mid-1960’s. It was conjectured by Rosa [13], and states that all trees are graceful. The graceful tree conjecture implies a very important result in graph theory. In particular, it implies Ringel’s conjecture [13], about the decomposition of complete graphs into trees. Even though a lot of work has been done on graceful and related labelings since the Graceful Tree Conjecture was introduced, no actual proof of the GTC was ever established. Relaxations of graceful labelings has been an approach to investigate such conjectures. There has only been scant progress toward the GTC, see [10], [14], and [16]. A recent approach to the conjecture is to relax some of the conditions on graceful labelings, [3] and [4]. Then one hopes to establish some cases of labelings under these relaxed conditions. A particular type of labeling under relaxed conditions are vertex- relaxed graceful labelings (VRG), which allows repeated vertex labels. These types of labelings have not been as extensively studied as other types of relaxed graceful labelings. It is not even clear what the diﬀerences between graceful graphs and vertex- relaxed graceful graphs are. As mentioned in [4], “in fact we have yet to encounter a connected ungraceful graph which is vertex-relaxed graceful.” In this thesis are generated some criteria that help to understand gracefulness 1 of graphs. We will be approaching this issue by relaxing the condition on vertex labels. We ﬁrst recall the gracefulness of paths, cycles, and complete graphs [14]. This will illustrate that the standard methods for studying gracefulness are ad hoc. One of the important class of graceful trees are caterpillars [13]. These are essentially formed from a path with edges attached to some of the vertices of the path. New classes of graceful graphs are often obtaine whereby one can construct new graceful graphs from existing graceful graphs. As a consequence, large classes of graceful trees can be constructed from trees that are known to be graceful, such as caterpillars. Computational methods, as in [14], give a way of investigating gracefulness of graphs. We will attempt to establish a more systematic approach, especially in regards to VRG labelings. In particular, we investigate necessary conditions for vertex-relaxed gracefulness in graphs. A goal is to establish easily computable methods to show that a particular labeling of a graph is not graceful. We will be using Faulhaber’s Formula to generate criterian that will help us determine if graphs are graceful or not. We generalize Faulhaber’s Formula to obtain a criterion to help us to decide if a given labeling is graceful or not. One of the main results is this thesis is Theoreom 4.5.1, a generalization of Faulhaber’s Formula. A consequence of this generalization is Proposition 5.0.2. This gives a criterion for vertex relaxed graceful labelings.

2 Chapter 2

Deﬁnitions and Conjectures

In this chapter, we will define relevant terms such as different types of graphs and different kinds of labelings. We will also review an important conjecture. Defini- tions follow from West, Introduction of Graph Theory [13]

2.1 Deﬁnitions

We begin by discussing the diﬀerent types of graphs that we will be using throughout this thesis. First, we will need to deﬁne a graph. Some of the graphs that we will be using are paths, cycles, complete graphs, trees, and caterpillars.

Deﬁnition 2.1.1. A graph G is a triple consisting of a vertex set V (G) which are nodes of the graph G, an edge set E(G) which connects some vertices of the graph G, and a relation that associates each edge with two vertices (not necessarily distinct) are called its endpoints. If u and v are adjacent then we write u ∼ v.

Deﬁnition 2.1.2. A simple graph is a graph having no loops or multiple edges.

Deﬁnition 2.1.3. A path is a simple graph whose vertices can be listed so that vertices are adjacent if and only if they are consecutive on the list.

3 Deﬁnition 2.1.4. A cycle is a simple graph whose vertices can be placed on a circle so that vertices are adjacent if and only if they appear consecutively on the circle.

Figure 2.1: Cycle graphs (C3 − C6)

Deﬁnition 2.1.5. A complete graph Kn is a simple graph in which each two vertices

n(n−1) are adjacent.A complete graph has 2 edges and n vertices, some examples are shown below[18]:

Figure 2.2: Kn Graphs [18]

Deﬁnition 2.1.6. A graph with no cycle is acyclic.A forest is an acyclic graph. A tree is a connected acyclic graph. 4 There are many diﬀerent kinds of trees: caterpillars,and paths are just a couple of examples.

Deﬁnition 2.1.7. A caterpillar is a tree with a single path containing at least one end point of every edge.

Below we have included an example of a caterpillar.

Figure 2.3: Example of a caterpillar graph

Deﬁnition 2.1.8. A graceful labeling of a graph G with m edges is an injective function f : V (G) → {0, . . . , m} such that distinct vertices receive distinct numbers and {|f(u) − f(v)| : uv ∈ E(G)} = {1, . . . , m}. A graph is graceful if it has a graceful labeling.

Note: Not all graphs are graceful.

Deﬁnition 2.1.9. A vertex relaxed graceful labeling (VRG) is a type of labeling as above under relaxed conditions which allow for repeated vertex labels. That is, the mapping of f : V (G) → {0, 1, ..., m} does not have to be injective.

An important conjecture of graceful labeling and graceful graphs is the Grace- ful Tree Conjecture.The Graceful Tree Conjecture was originated from Kotzig Ringel in 1964 [11].

Conjecture 2.1.10. Every tree has a graceful labeling.

5 We will show some properties of graceful graphs and the requirements for a graph to be graceful. As well as provide some examples of graceful graphs and non-graceful graphs.

7 6 5 0 3 2 2 8 0 4 4 1 4

1 2 3 3 0 1

Figure 2.4: Three examples of graceful graphs and their graceful labelings.

Figure 2.5: Two examples of non-graceful graphs.

Recall that a vertex-relaxed graceful labeling f of G can have repeated vertex labels. However, it is easy to see from the following proposition, that there are still signiﬁcant variations on the labelings of a vertex-relaxed graceful labeling.

Proposition 2.1.11. Let G be a graph with at least 1 edge. If f is a vertex-relaxed graceful labeling of G then the labels f(v) cannot all be even or odd.

Proof. As the edge set is labeled by {|f(u) − f(v)| : uv ∈ E(G) = {1, . . . , m}} then there is an edge uv with label 1. Thus |f(u)−f(v)| = 1 so f(v) and f(u) have opposite parity.

6 Chapter 3

Arithmetic Properties of Vertex-Relaxed Graceful Labelings

In this chapter we will use classical results to help us generate certain arithmetic properties for vertex relaxed graceful labelings. These arithmetic results can be used to obtain necessary conditions on a labeling f of a graph G for f to be vertex- relaxed graceful. Such conditions can be used to demonstrate that certain graphs do not have a vertex-relaxed graceful labeling. Also, such conditions can be used to restrict the number of possible labelings that are vertex-relaxed labeling. Thus such conditions can be used for computational methods to study possible labelings of small graphs. We will use the following well-known results. They are easily proven using induction.

Proposition 3.0.1.

n X n(n + 1) i k = 2 k=1 n X n(n + 1)(2n + 1) ii k2 = 6 k=1

An example of an application of these results is the following.

Proposition 3.0.2. Let G be a graph with m edges, and let f be a vertex-relaxed graceful labeling of G. Then

X m(m + 1) deg(v)f(v) ≡ mod 2. 2 v ∈ V (G) 7 Proof. As f : V (G) → {0, 1, . . . , m} is a vertex-relaxed graceful labeling, the map g : E(G) → {1, . . . , m} given by g(uv) = |f(u) − f(v)| for uv ∈ E(G), u, v ∈ V (G) is a bijection. Thus |f(u) − f(v)| ranges over {1, . . . , m} for u ∼ v. Thus we have

m X X m(m + 1) |f(u) − f(v)| = k = 2 u∼v k=1 u,v ∈V (G)

and also X X |f(u) − f(v)| ≡ f(u) + f(v) mod 2. u∼v u∼v u,v ∈V (G) u,v ∈V (G)

Note that in the sum X f(u) + f(v) u∼v u,v ∈V (G)

for very u ∈ V (G), the terms f(u) in the sum occur once for every vertex v ∈ V (G) that is incident with u. Thus each term f(u) will occur exactly deg (u) times. So we have X X f(u) + f(v) = deg (v)f(v). u∼v v∈V (G) u,v ∈V (G)

Therefore,

X X X deg (v)f(v) = f(u) + f(v) ≡ |f(u) − f(v)| mod 2 v∈V (G) u∼v u∼v u,v ∈V (G) u,v ∈V (G) m(m + 1) ≡ mod 2. 2

As a consequence, we can obtain conditions on vertex-relaxed gracefullness for certain classes of graphs.

Corollary 3.0.3. Let G have an Eulerian path from vertex u to vertex v, and let

8 |E(G)| = m. If f is a vertex-relaxed graceful labeling from u to v then

m(m + 1) f(u) + f(v) ≡ mod 2. 2

Proof. If G has an Eulerian path from u to v then by [Theorem 1.2.26 [13]], every vertex of G except u and v have even degree. Vertices u and v may have odd degree. Thus X deg (u)f(v) ≡ f(u) + f(v) mod 2 v∈V (G) and so m(m + 1) f(u) + f(v) ≡ mod 2 2

by Proposition 3.0.2.

We also have the following (known, see [17]) result

Corollary 3.0.4. If G has an Eulerian circuit and G is graceful with |E(G)| = m then m(m + 1) ≡ 0 mod 2. 2

Proof. As G has an Eulerian circuit, then by [Theorem 1.2.26 [13]] every vertex of G has even degree. Every graceful labeling is vertex-relaxed graceful, so by Proposition 3.0.2, X m(m + 1) 0 = deg (v)f(u) ≡ mod 2. 2 v∈V (G)

As a consequence, for example, an n − cycle with n ≡ 2 mod 4 cannot be graceful. That is because such a cycle would also have n = 4j + 2 edges, and

n(n + 1) (4j + 2)(4j + 3) ≡ = (2j + 1)(4j + 3) ≡ 1 mod 2. 2 2

9 For graphs with an Eulerian path from u to v with deg(u), deg(v) odd, any vertex- relaxed graceful labeling f must label u and v both even or both odd. This gives a restriction on such possible labelings. We now give some lemmas that will be useful in proving further results.

Lemma 3.0.5. Let f be a labeling of G. Then

X X f(u)k + f(v)k = deg(v)f(v)k. u∼v v∈V (G)

Proof. Note that:

X X f(u)k + f(v)k = f(u)k + f(v)k u∼v u,v∈E(G)

In the latter sum for every v ∈ V (G), f(v)k will occur once for each edge incident with v. Therefore f(u)k will occur deg(v) times, thus:

X X f(u)k + f(v)k = deg(v)f(v)k. u∼v v∈V (G)

Deﬁnition 3.0.6. Given a labeling f of G,

degj,n(V ) = #{w ∈ V (G)| w adjacent to v, |f(v) − f(w)| ≡ j mod n}

For example, for the modulo 2 case, degj,2 is the number of vertices u adjacent to v where f(u) − f(v) ≡ j mod 2. We have the following.

Lemma 3.0.7.

X 2 2 |f(u)−f(v)| X 2 (f(u) + f(v) )(−1) = (deg0,2(v) − deg1,2(v))f(v) u∼v v∈V (G) 10 Proof. We have

X (f(u)2 + f(v)2)(−1)|f(u)−f(v)| u∼v X X = (f(u)2 + f(v)2) − (f(u)2 + f(v)2) u∼v u∼v f(u),f(v) f(u),f(v) same parity opposite parity

X 2 X 2 = deg0,2(v)f(v) − deg1,2(v)f(v) v∈V (G) v∈V (G)

X 2 = (deg0,2(v) − deg1,2(v))f(v) v∈V (G)

To use these lemmas and obtain a new result about such labelings , we require another result about summing squares of integers.

Lemma 3.0.8. n X n(n + 1) (−1)kk2 = (−1)n 2 k=1 Proof. By induction on n. Clearly for n = 1 it holds. Assume it holds for n − 1 so then n−1 X (n − 1)n (−1)kk2 = (−1)n − 1 2 k=1 So by adding (−1)nn2 to both sides we get:

n X (n − 1)n (−1)kk2 = (−1)n−1 + (−1)n n2 2 k=1 (n − 1)n = (−1)n−1 − n2 2 (n − 1)n 2n2 = (−1)n−1 − 2 2 (n2 − n 2n2 = (−1)n−1 − 2 2

11 −n2 − n = (−1)n−1 2 n2 + n = (−1)n 2 n + 1 = n (−1)n. 2

The point for the following proposition is to give a criterion for a vertex-relaxed graceful labeling that can be checked at each vertex.

Proposition 3.0.9. Let f be a vertex-relaxed graceful labeling of a graph G with |E(G)| = m. Then

X 2 (deg(v) − deg0,2(v) + deg1,2(v))f(v) v∈V (G)

m(m + 1) 2m + 1 ≡ − (−1)m mod 8 2 3

Proof. As f is a vertex-relaxed graceful labeling, the value |f(u) − f(v)| ranges over {1, . . . , m} as u, v ∈ V (G) range over adjacent vertices. Then from Proposition 3.0.8,

m X X m(m + 1)(2m + 1) |f(u) − f(v)|2 = k2 = . 6 u,v∈V (G) k=1 u∼v

Thus we have

m(m + 1)(2m + 1) X = (f(u) − f(v))2 6 u∼v X X = f(u)2 + f(v)2 − 2 f(u)f(v) u∼v u∼v X X = deg(v)f(v)2 − 2 f(u)f(v) v∈V (G) u∼v 12 from Lemma 3.0.5. This can be written as

X X X deg(v)f(v)2 − 2 f(u)f(v) − 2 f(u)f(v). v∈V (G) u∼v u∼v f(u),f(v) f(u),f(v) same parity opposite parity

We also have from Lemma 3.0.7 and 3.0.8,

(−1)m m(m + 1) X = |f(u) − f(v)|2(−1)|f(u)−f(v)| 2 u∼v

X 2 X |f(u)−f(v)| = (deg0,2(v) − deg1,2(v))f(v) − 2 f(u)f(v)(−1) v∈V (G) u∼v

X 2 X = (deg0,2(v) − deg1,2(v))f(v) − 2 f(u)f(v) v∈V (G) u∼v f(u),f(v) same parity X + 2 f(u)f(v). u∼v f(u),f(v) opposite parity

Subtracting these, we obtain

m(m + 1)(2m + 1) (−1)mm(m + 1) − 6 2 X X = deg(v)f(v)2 − 2 f(u)f(v) v∈V (G) u,v ∈V (G) u∼v opposite parity

X 2 X − (deg0,2(v) − deg1,2(v)) f(v) − 2 f(u)f(v) v∈V (G) u∼v f(v),f(u) opposite parity

X 2 X = (deg(v) − deg0,2(v) + deg1,2(v)) f(v) − 4 f(u)f(v). v∈V (G) u,v∈V (G) u∼v f(u), f(v) opposite parity

13 Therefore,

X 2 X (deg(v) − deg0,2(v) + deg1,2(v)) f(v) − 4 f(u)f(v) v∈V (G) u,v∈V (G) u∼v f(u), f(v) opposite parity m(m + 1) 2m + 1 = − (−1)m . 2 3

As f(u) and f(v) have opposite parity, the product f(u)f(v) must be even. Thus taking the equation modulo 8 we get

X 2 (deg(v) − deg0,2(v) + deg1,2(v)) f(v) v∈V (G)

m(m + 1) 2m + 1 ≡ − (−1)m mod 8. 2 3

As an application of this result, we have the following.

Corollary 3.0.10. If G is a graph with m edges, 8 deg(v) for all v ∈ V (G) and there

is a vertex-relaxed graceful labeling such that deg0,2(v) = deg1,2(v) for all v ∈ V (G).

Then 8 m.

Proof. As deg(v) ≡ 0 mod 8 and deg0,2(v) = deg1,2(v) for all v ∈ V (G), then

X 2 (deg(v) − deg0,2(v) + deg1,2(v)) f(v) ≡ 0 mod 8. v∈V (G)

Thus by Proposition 3.0.9 we have

m(m + 1) 2m + 1 − (−1)m ≡ 0 mod 8. 2 3

P As v∈V (G) deg(v) = 2m ≡ 0 mod 8 then 2m ≡ 0 mod 8 so 4 m. Setting m = 4j 14 we get

4j(4j + 1) 2(4j) + 1 8j + 1 0 ≡ − (−1)4j = 2j(4j + 1) − 1 mod 8 2 3 3 8j − 2 = (8j + 2j) mod 8 3 ≡ (8j + 2j)(8j − 2)3 mod 8

≡ 4j mod 8.

Thus 8 4j and 8 m.

Such consequences give information about vertex-relaxed graceful labelings of graphs. We want to extend these results. However, to do so we will need further summation formulas. Faulhaber’s Formula gives such summations, and we discuss this result and give a generalization that we use.

15 Chapter 4

Faulhaber’s Formula and a Generalization

4.1 Prerequisites for Faulhaber’s Formula

We ﬁrst give an introduction to Bernoulli numbers and Bernoulli polynomials.

The Bernoulli numbers, denoted Bk, are deﬁned by the generating function:

∞ z X zk = B , |z| < 2π. ez − 1 k k! k=0

These numbers have been studied extensively and appear in numerous theorems and results in mathematics. The following are the Bernoulli numbers for k ∈ {0 − 6}:

−1 1 −1 1 B = 1,B = ,B = ,B = 0,B = ,B = 0,B = . 0 1 2 2 6 3 4 30 5 6 42

Bernoulli polynomials, denoted Bk(x), are deﬁned by the generating function

∞ zezx X zk = B (x) . ez − 1 k k! k=0

It is not obvious that these functions Bk(x) are polynomials, but note that:

∞ ! ∞ ! z X zk X (zx)k ezx = B ez − 1 k k! k! k=0 k=0

z2 z3 z2x2 z3x3 = B + B z + B + B + ... 1 + zx + + + ... 0 1 2 2! 3 3! 2! 3!

16 z2 = B + (B x + B )z + (B x2 + 2B x + B ) + ... 0 0 1 0 1 2 2

Which leads us to the ﬁrst few Bernoulli Polynomials:

1 1 B (x) = 1,B (x) = x − ,B (x) = x2 − x + 0 1 2 2 6

3 1 1 B (x) = x3 − x2 + x, B (x) = x4 − 2x3 + x2 − . 3 2 2 4 30 5x4 5x3 x 5x4 x2 1 B (x) = x5 + + − ,B (x) = x6 − 3x5 + − + . 5 2 3 6 6 2 2 42

Note that

∂ zezx zezx = z ∂x ez − 1 ez − 1 ∞ X zk = z B (x) k k! k=0 ∞ X zk+1 = B (x) k k! k=0 ∞ X zk = B (x) k−1 (k − 1)! k=1 ∞ X zk = kB (x) k−1 k! k=1 and ∞ ! ∞ ∂ X zk X ∂ zk B (x) = B (x) . ∂x k k! ∂x k k! k=0 k=1

∂ Therefore we see that ∂x Bk(x) = kBk−1(x). Therefore we can apply an induction proof to see that Bk(x) is a polynomial of degree k. As B0(x) = B0 the result holds for k = 0, now assume that Bk−1(x) is a polynomial of degree k − 1 then so is

∂ kBk−1(x) = ∂x Bk(x). Therefore, Bk(x) must be a polynomial of degree k.

17 4.2 Introduction to Faulhaber’s Formula

PN m Johann Faulhaber found explicit formulas for the sums n=1 n for m = 1, 2,..., 23. Jakob Bernoulli found the general formula, but it has been named after Faulhaber. The resulting theorem is stated as the following:

Theorem 4.2.1. Faulhaber’s Theorem Let m ≥ 1 be an integer, then

n X 1 km = (B (n + 1) − B (1)). m + 1 m+1 m+1 k=1

As an example, let m = 1, then we have the following

n X n(n + 1) n = 2 k=1

and Faulhaber’s formula gives

1 1 (B (n + 1) − B (1)) = (B (n + 1) − B (1)) 1 + 1 1+1 1+1 2 2 2 1 1 1 = ((n + 1)2 − (n + 1) + − 12 − 1 + 2 6 6 n(n + 1) = . 2

So we see that Theorem 4.2.1 gives the expected result.

4.3 Proof of Faulhaber’s Formula

Now we will proceed with a proof for Faulhaber’s Formula. The following proof is adapted from Aigner’s “A Course in Enumeration”, [Springer, 2007]. We

18 begin by recalling the generating function of the Bernoulli Numbers,

∞ x X xn = B . ex − 1 n n! n=0

As well as the generating function of the Bernoulli Polynomials,

n X n B (x) = B xk. n k n−k k=0

It is well-known that Bn(1) = Bn, and we have

∞ n ! n ∞ X X xm X X km xm km = m! m! m=0 k=1 k=1 m=0 n ∞ X X (kx)m = m! k=1 m=0 n X = ekx k=1 e(n+1) x − 1 = ex − 1 x (e(n+1) x − 1) = . (ex − 1) x

Now,

b e(n+1)x−1 P∞ ((n+1) x) − 1 = b=0 b! x x P∞ (n+1)b xb = b=1 b! x ∞ X (n + 1)b xb−1 = b! b=1 ∞ X (n + 1)b+1 xb = . (b + 1)! b=0

19 We obtain the following

∞ n ! X X xm x e(n+1) x − 1 km = m! ex − 1 x m=0 k=1 ∞ a ! ∞ b+1 b ! X Ba x X (n + 1) x = a! (b + 1)! a=0 b=0 ∞ b+1 ! X X Ba (n + 1) = xm a! (b + 1)! m=0 a+b=m ∞ b+1 ! m X X Ba (n + 1) x = m! . a! (b + 1)! m! m=0 a+b=m

By setting the mth coeﬃcients equal we get

n b+1 X X Ba (n + 1) km = m! a! (b + 1)! k=1 a+b=m m j+1 X Bm−j (n + 1) = m! (m − j)! (j + 1)! j=0 m j+1 1 X Bm+1−(j+1) (n + 1) = (m + 1)! m + 1 (m + 1 − (j + 1))! (j + 1)! j=0 m 1 X m + 1 = B (n + 1)j+1 m + 1 m+1−(j+1) j + 1 j=0 m+1 1 X m + 1 = B (n + 1)j m + 1 m+1−j j j=1 m+1 ! 1 X m + 1 = B (n + 1)j − B . m + 1 m+1−j j m+1 j=0

Therefore, rewriting the right-hand-side above we get

n X 1 km = (B (n + 1) − B (1)) . m + 1 m+1 m+1 k=1

20 4.4 Introduction to an Extension of Faulhaber’s Formula

In order to proceed with the extension of Faulhaber’s formula we review some known results, which leads us to the next topic. It is an expression obtained by Euler for the alternating sum: n X (−1)n km. n=1

The expression involves a sequence of integers {G2n}n≤1 called Genocchi Numbers [9]. These numbers are related to Bernoulli numbers by the following expression see [8]:

2n G2n := 2(2 − 1)B2n , n ≥ 1.

The following theorem gives a variation of Faulhaber’s formula. Let {G2n}n≥1 be the sequence of Genocchi Numbers, then

n 2m m X n X 2m G2k (−1)nk2m = (−1)n + (−1)m+k+1 n2m−2k+1 2 2k − 1 4k n=1 k=1 n X (−1)nk2m+1 = n=1 2m+1 m+1 n X 2m + 1 G2k G2m+2 (−1)n + (−1)m+k+1 n2m+1−2k+1 + (−1)m+1 2 2k − 1 4k 4(m + 1) k=1

As a variation of Faulhaber’s Formula, similarly we can use this result to obtain arithmetic conditions for vertex-relaxed graceful labelings. However, we can also generalize this result. The ﬁrst tool that we will be using to generate Faulhaber’s Formula will be the rearrangement of series. As a ﬁrst step we establish the following classical result.

Theorem 4.4.1. Let {bn}n≥0 be a rearrangement of {an}n≥0 where an ≥ 0 for every

21 n ≥ 0. Then ∞ ∞ X X an = bn. n=0 n=0 PN PN Proof. Let An = n=0 an and Bn = n=0 bn, and set limN→∞ An = A and

0 limN→∞ Bn = B. Note that A ≥ AN and B ≥ BN for any N, for an N let N be large enough so that

{a1, . . . , aN } ⊆ {b1, . . . , bN 0 }.

Then we have 0 N N X X AN = an ≤ bn = BN 0 ≤ B. n=0 n=0

Thus AN ≤ B and in a similar way we get BN ≤ A for any N. Taking the limits as N → ∞ we get:

A = lim An ≤ Bn and B = lim Bn ≤ An. N→∞ N→∞

So A ≤ B and B ≤ A so A = B and we get the desired result:

∞ ∞ X X an = bn. n=0 n=0

Now we can establish a main tool that will help us to generalize Faulhaber’s Theorem.

P∞ Theorem 4.4.2. Let n=0 an be an absolutely convergent series. If {bn}n≥0 is a P∞ rearrangement of {an}n≥0 then n=0 bn is absolutely convergent and

∞ ∞ X X an = bn. n=0 n=0

P∞ P∞ Proof. As n=0 an is absolutely convergent, we have n=0 |an| < ∞ and from the

22 previous result, ∞ ∞ X X |an| = |bn|. n=0 n=0

Now , bn = |bn| − (|bn| − bn) and |bn| ≥ 0 and |bn| − bn ≥ 0. Also, note that

|bn| − bn = 2|bn| or 0, therefore:

N N X X bn = |bn| − (|bn| − bn) n=0 n=0 N N X X = |bn| − |bn| − bn. n=0 n=0

PN PN PN 0 Note that n=0 |bn| is absolutely convergent, and n=0 |bn| − bn = 2 n=0 |bn| where

0 |bn| = |bn| or 0 and so this sum is also absolutely convergent. As {|an|}n≥0 is a rearrangement of {|bn|}n≥0 and {|an| − an}n≥0 is a rearrangement of {|bn| − bn}n≥0 then from the previous result we get:

N N N X X X |an| = |bn| = lim |bn| N→∞ n=0 n=0 n=0 and N N N X X X |an| − an = |bn| − bn = lim |bn| − bn. N→∞ n=0 n=0 n=0 Thus

∞ N N N X X X X an = lim an = lim |an| − |an| − an N→∞ N→∞ n=0 n=0 n=0 n=0 ∞ ∞ X X = |an| − |an| − an n=0 n=0 ∞ ∞ X X = |bn| − |bn| − bn n=0 n=0 N N X X = lim |bn| − |bn| − bn N→∞ n=0 n=0

23 N X = lim bn N→∞ n=0 N X = bn. n=0

P∞ P∞ P∞ So n=0 bn is absolutely convergent and n=0 bn = n=0 an.

These theorems are classical, but the proofs here were taken from the lecture notes of Oleg Zabornski[19].

4.5 A Generalization of Faulhaber’s Formula

In this section we will be working on producing a more generalized version of Faulhaber’s Formula. This particular generalization is a main result of the thesis and can used to help us ﬁnd some further criteria for vertex relaxed-graceful labelings. We will use some of the previous results.

For n ∈ Z and r ∈ Z>0, the least non-negative residue of n modulo r is the unique integer nr so that 0 ≤ nr ≤ r − 1 and n ≡ nr mod r.

r Theorem 4.5.1. Let ξr ∈ C so that (ξr) = 1 and for n ∈ Z let nr be the least non-negative residue of n modulo r. Then we have

n m r−1 X r X j + n + 1 j + (n + 1)r ξk km = ξj+n+1 B − B . r m + 1 r m+1 r m+1 r k=0 j=0

P∞ k m Proof. If |x| < 1 then k=0 x k is absolutely convergent. This can be shown by using the ratio test, which says that

∞ ak+1 X If lim = L < 1 then ak is absolutely convergent. k→∞ ak k=0

24 k m Thus for ak = x k we have

k+1 m ak+1 x (k + 1) lim = lim k m k→∞ ak k→∞ x k k + 1m = |x| lim k→∞ k k + 1m = |x| lim k→∞ k = |x| 1m = |x| < 1.

P∞ k m Thus by Theorem 4.4.2 we can rearrange the series k=0 x k and obtain the same limit. So we have

∞ ∞ ∞ ∞ X X X X xk km − xk+n+1(k + n + 1)m = xk km − xk km k=0 k=0 k=0 k=n+1 n ∞ ∞ X X X = xk km + xk km − xk km k=0 k=n+1 k=n+1 n ∞ X X = xk km + (xk km − xk km) k=0 k=n+1 n X = xk km. k=0

Note that the rearrangement occurred in the second to last line. Let ξr ∈ C so that r P∞ k m (ξr) = 1 and let be small so that 0 < < 1. Then k=0(ξr) k is absolutely convergent. Thus, from the previous argument we have

n ∞ ∞ X k m X k m X k+n+1 m (ξr) k = (ξr) k − (ξr) (k + n + 1) . k=0 k=0 k=0

Now any absolutely convergent sum can be rearranged by taking, for any integer r > 0, the parts of 0 modulo r, then 1 modulo r, up to r − 1 modulo r. That is we

25 can rearrange the sum to be

∞ r−1 ∞ X X X ak = ak. k=0 j=0 k=0 k ≡ j mod r

Thus we have ∞ r−1 ∞ X k m X X k m (ξr) k = (ξr) k k=0 j=0 k=0 k ≡ n+1+j mod r and ∞ r−1 ∞ X k+n+1 m X X k+n+1 m (ξr) (k + n + 1) = (ξr) (k + n + 1) . k=0 j=0 k=0 k ≡ j mod r So we have ∞ ∞ X k m X k+n+1 m (ξr) k − (ξr) (k + n + 1) k=0 k=0   r−1 ∞ ∞ X  X k m X k+n+1 m =  (ξr) k − (ξr) (k + n + 1)  . j=0 k=0 k=0 k ≡ n+1+j mod r k ≡ j mod r

k n+1+j r Factoring out ξr = ξr as (ξr) = 1, we have

  r ∞ ∞ X X X ξj+n+1  kkm − k+n+1(k + n + 1)m r   j=0 k=0 k=0 k≡n+1+j mod r k≡j mod r

 n+1−(n+1)r  r r −1 X j+n+1 X rl+j+(n+1)r m = ξr  (rl + j + (n + 1)r)  j=0 l=0 where (n + 1)r ≡ (n + 1) mod r and 0 ≤ (n + 1)r ≤ r − 1. Thus,

n+1−(n+1)r n r−1 r −1 X k m X j+n+1 X rl+j+(n+1)r m (ξr) k = ξr (rl + j + (n + 1)r) . k=0 j=0 l=0

Note that as (n + 1)r ≡ (n + 1) mod r then (n + 1) − (n + 1)r ≡ 0 mod r so

26 r ((n + 1) − (n + 1)r). As the sums above are all ﬁnite, then we can take = 1 and get the equation

n+1−(n+1)r n r−1 r −1 X k m X j+n+1 X m ξr k = ξr (rl + j + (n + 1)r) . k=0 j=0 l=0

We now apply the following generalization of Faulhaber’s formula, which is (1) from [6]: M−1 X ym x x (yl + x)m = B M + − B . m + 1 m+1 y m+1 y l=0 This formula gives

n+1−(n+1)r n r−1 r −1 X k m X j+n+1 X m ξr k = ξr (rl + j + (n + 1)r) k=0 j=0 l=0 r−1 m X r n + 1 − (n + 1)r j + (n + 1)r = ξj+n+1 B + − r m + 1 m+1 r r j=0 j + (n+) B r m+1 r m r−1 r X j + n + 1 j + (n + 1)r = ξn+1 ξj B − B . m + 1 r r m+1 r m+1 r j=0

This gives the result.

27 Here are some examples of Theorem 4.5.1

2πi 1) With the following given conditions: ξ3 = e 3 , r = 3, m = 2, n ≡ 0 mod 3

n " 2 # X 32 X j + n + 1 j + 1 ξk k2 = ξn+1 ξj B − B 3 3 3 3 3 3 3 3 k=1 j=0 1 √ = n(4 + (3 − i 3)n). 6

2πi 2) With the following given conditions: ξ3 = e 3 , r = 3, m = 4, n ≡ 0 mod 3

n X 1 √ √ ξk k4 = n(−8 + 4i 3n + 8n2 + (3 − i 3)n3) 3 6 k=1

2πi 3) With the following given conditions: ξ4 = e 4 = i, r = 4, m = 2, n ≡ 0 mod 4

n X n2 in2 ik k2 = n + − . 2 2 k=1

2πi 4) With the following given conditions: ξ4 = e 4 = i, r = 4, m = 4, n ≡ 0 mod 4

n X n4 n4 ik k4 = −4n + 2n3 + + i 3n2 − . 2 2 k=1

28 Chapter 5

A Further Criterion for Vertex-Relaxed Graceful Labelings

Now that we have established a generalized version of Faulhaber’s Formula, we will be using it to establish a new criterion for vertex-relaxed graceful labelings of graphs. We begin with the following lemma.

Lemma 5.0.1. Let f : V (G) → Z≥0 be a labeling of the vertex of a graph G, and

r let ξr ∈ C so that (ξr) = 1. Then

r−1 ! X k k |f(u)−f(v)| X X j 2 f(u) + f(v) ξr = ξr degj,r(v) f(v) u∼v v∈V (G) j=0 u,v ∈V (G)

where

degj,r(v) = #{w ∈ V (G)|w ∼ v, |f(v) − f(w)| ≡ j mod r}.

Proof. We have

Now, in the inner sum we have

X X f(u)k + f(v)k = f(u)k + f(v)k. u∼v uv∈E(G) |f(u)−f(v)|≡j mod r |f(u)−f(v)|≡j mod r

29 In the latter sum, for v ∈ V (G), f(v)k will occur each time there is an edge uv incident with v and so that the labels f(u) and f(v) have the property that |f(u) − f(v)| ≡ j mod r. It follows that we have

X k k X k f(u) + f(v) = degj,r(v)f(v) . uv∈E(G) v∈V (G) |f(u)−f(v)|≡j mod r

Thus

r−1 X k k |f(u)−f(v)| X j X k k f(u) + f(v) ξr = ξr f(u) + f(v) u∼v j=0 u∼v u,v∈V (G) |f(u)−f(v)|≡j mod r r−1 X j X k = ξr degj,r(v)f(v) j=0 v∈V (G) r−1 ! X X j k = ξr degj,r(v) f(v) . v∈V (G) j=0

This lemma together with the generalization of Faulhaber’s Theorem applied

2πi to r = 3 and ξ3 = e 3 , gives the following main result for a vertex-relaxed graceful abeling of a graph.

Proposition 5.0.2. Let m ≡ 0 mod 3 and let f be a vertex-relaxed graceful labeling of G. Then X m(m + 3) deg (v)f(v) ≡ mod 2 0,3 2 v∈V (G) where

deg0,3(V ) = #{w ∈ V (G)|w ∼ v, |f(v) − f(u)| ≡ 0 mod 3}.

This proposition is the main criterion that we use to study vertex-relaxed graceful labelings.

√ 2πi −1 i 3 3 3 Proof. Let m ≡ 0 mod 3 and let ξ = e = 2 + 2 . Then (ξ3) = 1, and taking 30 r = 3 in the previous theorem we get

m 2 " 2 # X 3 X j + m + 1 j + (m + 1)3 ξkm2 = ξj+m+1 B − B 3 2 3 3 3 3 3 k=1 j=0 m + 1 1 m + 2 = 3 ξ B − B + ξ2 B 3 3 3 3 3 3 3 3 2 m − B + B + 1 − B (1) 3 3 3 3 3 √ 2m m2 i 3 m2 = + − . 3 2 6

√ 4πi −1 i 3 2 3 Now let’s apply the theorem, but with ξ3 = e = 2 − 2 in place of ξ3. We are 2 6 able to do this because (ξ3 ) = ξ3 = 1. But note that, for x the complex conjugate at

2 x, we have ξ3 = ξ3. Therefore

√ m m m 2 X k X k X k 2 m i 3 ξ2 m2 = ξ m2 = (ξ ) m2 = m + − m2 3 3 3 3 3 6 k=1 k=1 k=1 √ 2 m2 i 3 = m + + m2. 3 3 6

Also recall that

m X m(m + 1)(2m + 1) m m2 m3 m2 = = + + . 6 6 2 3 k=1

On the other hand, for f a vertec-relaxed graceful labeling of G,

m X k 2 X 2 |f(u)−f(v)| ξ3 k = |f(u) − f(v)| ξ3 k=1 u∼v u,v∈V (G)

X 2 2 |f(u)−f(v)| X |f(u)−f(v)| = f(u) + f(v) ξ3 − 2 f(u)f(v) ξ3 . u∼v u∼v u,v∈V (G) u,v∈V (G)

31 By previous the lemma this is

X 2 2 X deg0,3(v) + ξ3 deg1,3(v) + ξ3 deg2,3(v) f(v) − 2 f(u)f(v) v∈V (G) u∼v |f(u)−f(v)|≡0 mod 3

X 2 X − 2 ξ3 f(u)f(v) − 2 ξ3 f(u)f(v). u∼v u∼v |f(u)−f(v)|≡1 mod 3 |f(u)−f(v)|≡2 mod 3

Therefore we have

X 2 2 X deg0,3(v) + ξ3 deg1,3(v) + ξ3 deg2,3(v) f(v) − 2 f(u)f(v) v∈V (G) u∼v |f(u)−f(v)|≡0 mod 3

X 2 X − 2 ξ3 f(u)f(v) − 2 ξ3 f(u)f(v) u∼v u∼v |f(u)−f(v)|≡1 mod 3 |f(u)−f(v)|≡2 mod 3 √ 2 1 i 3 = m + m2 − m2. 3 2 6

2 Using ξ3 instead of ξ3 we get, by the same argument,

X 2 2 X deg0,3(v) + ξ3 deg1,3(v) + ξ3 deg2,3(v) f(v) − 2 f(u)f(v) v∈V (G) u∼v |f(u)−f(v)|≡0 mod 3

2 X X − 2 ξ3 f(u)f(v) − 2 ξ3 f(u)f(v) u∼v u∼v |f(u)−f(v)|≡1 mod 3 |f(u)−f(v)|≡2 mod 3 √ 2 1 i 3 = m + m2 + m2 3 2 6

and taking ξ3 = 1 we get

32 Now we add up the three equations. Note that

√ ! √ ! −1 i 3 −1 i 3 1 + ξ + ξ2 = 1 + + + − = 0. 3 3 2 2 2 2

Thus we get

X X 4 m m2 m3 3 deg (v)f(v)2 − 6 f(u)f(v) = m + m2 + + + 0,3 3 6 2 3 v∈V (G) u∼v |f(u)−f(v)|≡0 mod 3 3 3 m3 = m + m2 + . 2 2 3

m3 As m ≡ 0 mod 3 then 9 ∈ Z so we can divide by 3 and we get

X X m(m + 1) m3 deg (v)f(v)2 − 2 f(u)f(v) = + . 0,3 2 9 v∈V (G) u∼v |f(u)−f(v)|≡0 mod 3

Finally, by taking the previous equation modulo 2 will give us

X m(m + 1) m3 deg (v)f(v) ≡ + mod 2. 0,3 2 9 v∈V (G)

As 9 ≡ 1 mod 2 and m3 ≡ m mod 2 then

m(m + 1) m3 m(m + 3) + ≡ mod 2. 2 9 2

Note that

  m(m + 3) 0 mod 2, if m ≡ 0 or 1 mod 4 ≡ 2  1 mod 2 if m ≡ 2 or 3 mod 4.

We know that for f a vertex-relaxed graceful labeling, that not all the labels 33 f(v) can be even or odd. As a simple consequence of this proposition, it follows that if m ≡ 1 or 3 mod 4 then the right-hand-side of the equation from Ptoposition 5.0.2 is 1 mod 2. Therefore for every odd label f(v) we cannot have deg0,2(v) even. As a consequence of the above result, for f a vertex-relaxed graceful labeling of G, for

such m, there must be an odd label f(v) with deg0,2(v) also odd.

5.1 An Application of Proposition 5.0.2

Next we apply Proposition 5.0.2 to a few examples. Graph of Figure 5.1 was taken from [15]. We start with restating Proposition 5.0.2. Let m ≡ 0 mod 3 and let f be a vertex-relaxed graceful labeling of G. Then

X m(m + 3) deg (v)f(v) ≡ mod 2 0,3 2 v∈V (G)

where

deg0,3(V ) = #{w ∈ V (G)|w ∼ v, |f(v) − f(u)| ≡ 0 mod 3}

0 4

Figure 5.1: Graceful Labeling Example 1[15]

34 The above graph has a graceful labeling with m = 6 edges. Notice that the vertex label f(v) = 1, an odd label, has exactly one vertex u adjacent to it so that |f(u) − f(v)| ≡ 0 mod 3. This gives us the following

X deg0,3(v)f(v) = 1(1) ≡ 1 mod 2. v∈V (G)

Looking at the right-hand side of Proposition 5.0.2, we get

m(m + 3) mod 2 ≡ 1 mod 2. 2

1 6 0

Figure 5.2: Graceful Labeling Example 2[15]

In the above graph we have a graceful labeling of a tree that is not a caterpillar. The following tree contains m = 6 edges. There exists a vertex v an odd label, f(v) = 5, so that there are an odd number of adjacent vertices u so that |f(v) − f(u)| ≡ 0 mod 3. By looking at the graph we get the following

X deg0,3(v)f(v) = 1(1) ≡ 1 mod 2. v∈V (G)

By looking at the right-hand side of Proposition 5.0.2 we have

m(m + 3) mod 2 ≡ 1 mod 2. 2

35 Thus we see that the criterion of Proposition 5.0.2 is veriﬁed. The next example is not easy to determine if the labeling is a vertex-relaxed graceful labeling.

0 1 4

6 20

22 24

10 16

8 27 14

Figure 5.3: Graceful Labeling Example 2[15]

The above graph has a labeling with m = 45 edges. Notice that the vertex labeled f(v) = 1, an odd label, has exactly one vertex u adjacent to it so that |f(u) − f(v)| ≡ 0 mod 3 namely f(u) = 16. The vertex labeled f(v) = 27, an odd label, has exactly one vertex u adjacent to it so that |f(u)−f(v)| ≡ 0 mod 3 namely f(u) = 0. Note that 1 and 27 are the only odd labels. Thus we have the following

X deg0,3(v)f(v) = 1(1) + 1(27) ≡ 0 mod 2. v∈V (G)

Let us look at the right-hand side of Proposition 5.0.2,

m(m + 3) 45(45 + 3) mod 2 ≡ mod 2 ≡ 1 mod 2. 2 2

36 It follows that the labeling is not a vertex-relaxed graceful labeling.

2 6

0 1

Figure 5.4: Graceful Labeling Example 2[15]

The above graph has a labeling with m = 12 edges. Notice that the vertex label f(v) = 9, an odd label, has exactly 2 vertices u adjacent to it so that |f(u)−f(v)| ≡ 0 mod 3. The other odd labeled,f(u) = 1, has no such vertices adjacent to it. This gives us the following

X deg0,3(v)f(v) = 2(9) ≡ 0 mod 2. v∈V (G)

Let us look at the right-hand side of Proposition 5.0.2

m(m + 3) 12(12 + 3) mod 2 ≡ mod 2 ≡ 0 mod 2. 2 2

Thus we see that the criterion is satisﬁed. Therefore it is likely that the labeling is vertex-relaxed graceful. Indeed, it is in fact a graceful labling.

37 Chapter 6

Conclusion

For G a graph, there are numerous different labelings. For a small graph it is not a big problem to list the possible labelings. However, as graphs get larger and larger, the number of labelings increases dramatically therefore it becomes increas- ingly harder to find a graceful labeling. For this reason it can be helpful to generate some criteria for graceful labelings. For example, a graceful labeling cannot have the same parity for all vertices. We also have the congruence condition from Rosa. He stated and proved Corollary 3.0.4. This criteria helps us to better understand gracefulness. By using similar reasoning and Faulhaber’s Formula we are able to obtain further criteria We are able to generate an extension of Faulhaber’s Formula. As a consequence we were able to create a new criteria such as Proposition 5.0.2. This occurs because unnecessary labelings are being eliminated, giving us stricter conditions on where and how the labelings are placed. We are able to put several conditions on the labelings and reduce the number of acceptable labelings. We applied such conditions in the examples at the end of chapter 5. Proposition 5.0.2 can be useful in more quickly determining if a particular labeling is not a vertex-relaxed labeling. Potentially, this and similar results can be integrated into different algorithms to help find vertex-relaxed graceful labelings of graphs.

38 Bibliography

[1] B.P Acharya, Construction of certain inﬁnite families of graceful graphs from a given graceful graph, Defence Science Journal, 32 no. 3, (1982),231-236.

[2] M. Aigner, A Course in Enumeration Grad. Texts in Math. 238, Springer, Berlin, 2007.

[3] J.C Bermond, Graceful graphs, radio antennae and French windmills, in graph theory and combinatorics(Proc.Conf. Open Univ., Milton Keynes, 1978), Res. Notes in Math., 34, Pitman, Boston, MA-London, 1979, 18-37.

[4] F. Van Bussel, Relaxed graceful labelings of trees, Elect. J. of Comb., 9, (2002), R4.

[5] F.R.K Chung, Labelings of graphs, Selected Topics in Graph Theory 3, Academic Press, San Diego, CA, 1988, 151-168.

[6] Giuseppe Dattoli, Clemente Cesarano, Silveria Lorenzutta, Bernoulli Numbers and Polynomials from a more general point of view (English, Italian summary) rend. Mat. Appl. 22 (2003), 193-202.

[7] J. Gallian, a dynamic survey of graph labelings, Electron. J. Combin. 5 (1998), Dynamic Survey 6, 43pp.

[8] Dominique Foata, Eulerian Polynomials: from Euler’s time to present(English Summary), the Legacy of Alladi Ramakrishnan in the Mathematical Science, Springer, New York(2010), 253-273.

39 [9] A.Genocchi, Intorno all’e spressione generale de’numeri Bernulliani, Aun. Sci. Mat. Fis. 3(1852), 395-405.

[10] P. Hunyiar and A. Haviar, All trees of diameter ﬁve are graceful, Discrete Math., 233, (2001), 133-150.

[11] G. Ringel, Problem 25 in Theory of Graphs and its Application, Proc. Symp. Smolenice 1963, (1964),162.

[12] A. Rosa, On certain valuations of vertices of a graph, in Theory of Graphs, international Symposium, Rome, July 1966, (1967), 349-355.

[13] D.B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, Inc., Upper Saddle River, NJ, 2001.

[14] R.M. Zhou, Graceful Labelings of Graphs, M.S. Thesis, COPPE/UFRJ, Systems of Engineering and Computer Sciences, Dec. 2016

[15] T.A. Redl, graceful graphs and graceful labelings: two mathematical programing formulas and some other results. Procedings of the Thirty - Fourth Southeastern International Conference Combinations, Graph Theory, and Computing, Congr. Numer., 164 (2003), 17-31.

[16] S.L. Zhou, All treed of diameter four are graceful, Annals of New York Academy of Science, 576 no.1,(1989),700-706.

[17] https://brilliant.org/wiki/sum-of-n-n2-or-n3/.

[18] https://math.stackexchange.com/questions/1129911/determine-the-number-of- paths-of-length-2-in-a-complete-graph-of-n-nodes.

[19] https://warwick.ac.uk/fc/sci/maths/people/staﬀ/oleg- zaboronski/analysis/notes-ma131-ch10.pdf.