Download Fibonacci numbers of Tadpole graph and more Thesis Mathematics in PDF only on Docsity!
Abstract
Fibonacci number of a tadpole graph: An Exposition
by John Victor L. Buscas
This paper is an exposition of J. DeMaio and J. Jacobson on “Fibonacci number of the tadpole graph”. Here, we study in detail. In [9] H.Prodinger and R.Tichy classified the Fibonacci number of different types of graphs such as path graph, cycle graph, star graph, wheel graph, ladder graph, and generalized Petersen graph. In this paper we introduce the formulae and identities for the Fibonacci number of Tadpole graph Tn,k using algebraic and combinatorial methods.
Table of Contents
Abstract ii
Acknowledgements iii
1 Introduction 1
2 Preliminaries and Notations 2
3 Main Concepts 6 3.1 Fibonacci and Lucas numbers............................ 6 3.2 The Tadpole Graph................................. 11 3.3 Tadpole Triangle................................... 27 3.4 The L-shaped Graph................................. 42
4 Summary and Recommendations 48 4.1 Summary....................................... 48 4.2 Recommendations.................................. 48
List of References 49
v
Chapter 2
Preliminaries and Notations
In this chapter we start with formal definitions with notations in able to understand the main concept of this paper.
Basic Concepts about set theory
Definition 2.1. A set consists of elements. Notation: x ∈ M means that x is an element of a set M (belongs to M ).[12].
Definition 2.2. A set A is a subset of a set B (A ⊂ B) if each element of A is also an element of B. In this case B is called a superset of A.[12].
Definition 2.3. Two sets A and B are equal (A = B) if they consist of the same elements (i.e., if A ⊂ B and B ⊂ A).[12].
Definition 2.4. If A is a subset of B and A 6 = B, then A is called a proper subset of B (notation: A ⊆ B).[12].
Definition 2.5. The empty set ∅ (called also the null set) contains no elements. It is a subset of any set.[12].
Definition 2.6. The intersection A ∩ B of two sets A and B consists of all elements that belong both to A and to B: A ∩ B = {x|x ∈ A and x ∈ B}.[12].
Definition 2.7. The union A ∪ B consists of all elements of A and all elements of B (and no other elements): A ∪ B = {x|x ∈ A or x ∈ B}.[12].
2
Preliminaries and Notations 3
Definition 2.8. The set difference A \ B consists of elements of A that are not elements of B: A \ B = {x|x ∈ A and x /∈ B}.[12].
Some Definitions with Notations about Graph theory
Definition 2.9. A graph G is a collection of points called vertices and lines called edges where two vertices are joined by an edge if they are related in some way.[1].
Definition 2.10. The path graph Pn composed of the vertex set V = { 1 , 2 , ..., n} and the edge set E = {{ 1 , 2 }, { 2 , 3 }, .., {n − 1 , n}}.[5].
Definition 2.11. The cycle graph Cn, is a path graph Pn with the additional edge { 1 , n}.[5].
Definition 2.12. Given a graph G, a subset S ⊆ V is an independent set of vertices if no two vertices in S are adjacent.[5].
Definition 2.13. Let i(G) =| I(G) | be the cardinality of the set of all independent sets in graph G.[5].
Definition 2.14. Let I(G) be the set of all independent in graph G.[5]
Definition 2.15. A collection of edges, no two adjacent, in a bipartite graph G is called a matching.[1].
Definition 2.16. Two graphs G and H are isomorphic graphs if the vertices of G can be relabeled to produce H. If G and H are isomorphic, then this is indicated by writing G ∼= H.[1].
Preliminaries and Notations 5
Definition 2.21. Let S be a sequence of numbers. A recurrence relation on S is a formula that relates all, but finite number of terms of S, to previous terms of S.[11].
Example 2.1. For Fibonacci sequence
The Fibonacci sequence is defined by relation FK = FK− 2 + FK− 1 , K ≥ 2 where F 0 = 0 and F 1 = 1 are called initial conditions Note: The recurrence relations also called difference equations.
Definition 2.22. Facts about Limits Involving Infinity.[13].
Let f be a function that is defined for all x ≥ a, for some number a. If as x takes larger and larger positive values, increasing without bound, the corresponding values of f (x) get very close (and possibly equal) to the real number L and the values of f (x) can be made arbitrarily close (as close as you want) to L for all large enough values of xn, then we say that the limit of f (x) as x approaches infinity is L, which is written,
x→lim+∞ f^ (x) =^ L
Similarly, if f is defined for all x ≤ b, for some number b, then
x→−∞lim f^ (x) =^ L
which is read the limit of f (x) as x approaches negative infinity is L, means that as x takes smaller and smaller negative values, decreasing without bound, the corresponding values of f (x) get very close (and possibly equal) to the real number L, and the values of f (x) can be made arbitrarily close (as close as you want) to L for all small enough values of x.
Chapter 3
Main Concepts
In this chapter, we present the main concept of this paper and we will prove some of its theorems.
3.1 Fibonacci and Lucas numbers
The Fibonacci number is series of numbers in which each number is the sum of the two preceding numbers starting from 0, 1, 1, 2, 3, 5, 8,...
Fibonacci number is denoted by Fn and Fibonacci sequence is defined recursively,
F 0 = 0 and F 1 = 1 ←− Initial conditions Fn = Fn− 1 + Fn− 2 , ∀n ≥ 2 ←− Recurrence relation Table 3.1: First 16 values of the Fibonacci numbers from n 0 1 2 3 4 5 6 7 8 9 10 11 12 n = 0 13 , 1 , 2 , .., 14 15 are: 15 Fn 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
And the Lucas number is series of numbers in which each number is the sum of the two preceding numbers starting from 2, 1, 3, 4, 7, 11,...
Lucas number is denoted by Ln and Lucas sequence is defined recursively,
L 0 = 2 and L 1 = 1 ←− Initial conditions Ln = Ln− 1 + Ln− 2 , ∀n ≥ 2 ←− Recurrence relation
n Table 3.2: First 16 values of the Lucas numbers from 0 1 2 3 4 5 6 7 8 9 10 11 n^ = 0 12 ,^1 ,^132 , .., 15 are: 14 15 Ln 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364
6
Main Concepts 8
The initial set of independent set by means of path graph i(Pn) = Fn+2 and for cycle graph i(Cn) = Ln is illustrated in Figure 3.1 and Figure 3.2. Figure 3.1: The set of all independent sets of P 1 ,P 2 ,P 3 ,P 4 and P 5
Figure 3.2: The set of all independent sets of C 3 ,C 4 and C 5
In [3] J.Dmaio and J.Jacobson used path graph and cycle graph in able to derive an identities related to Fibonacci and Lucas numbers. In Table 3.3 and Table 3.4 shows the initial values of the number of independent sets in path graph Pn is the Fibonacci number Fn+2 and number of independent sets in Cn is the Lucas number Ln.
Main Concepts 9
n 1 2 3 4 Table 3.3: Initial values of 5 6 7 8 9 i (Pn) = 10 Fn 11 +2 12 13 14 15 i^ F(nP+2^ F^3 F^4 F^5 F^6 F^7 F^8 F^9 F^10 F^11 F^12 F^13 F^14 F^15 F^16 F^17 n)^2 3 5 8 13 21 34 55 89 144 223 377 610 987 Table 3.4: Initial values of i(Cn) = Ln L^ n^3 4 5 6 7 8 9 10 11 12 13 14 i(Cnn^ ) L 43 L 74 L 115 L 186 L 297 L 478 L 769 L 12310 L 19911 L 32212 L 52113 L 84314 1364 L^15
In our Illustrations, we denote the membership of independent set S in G by shading the vertices in S.
Example 3.1. Using Table 3.3 and Table 3.4.
Consider three vertices in P 3 and we will find the number of independent sets by Table 3. we have, i(P 3 ) = F3+2 = F 5 = 5.
I(P 3 ) = {∅, {p 1 }, {p 2 }, {p 3 }, {p 1 , p 3 }}.
Consider three vertices in C 3 and we will find the number of independent sets by Table 3. we have, i(C 3 ) = L 3 = 4.
I(C 3 ) = {∅, {c 1 }, {c 2 }, {c 3 }}.
Main Concepts 11
The Gibonacci number is denoted by Gn and Gibonacci sequence is defined recursively, G 1 = a and G 2 = b ←− Initial conditions Gn = Gn− 1 + Gn− 2 , n ≥ 3 ←− Recurrence relation n G Table 3.5: Initial representation of Gibonacci numbers G^1 G^2 G^3 G^4 G^5 G^6 G^7 G^8 G^9 G^10 n a^ b^ a^ +^ b^ a^ + 2b^2 a^ + 3b^3 a^ + 5b^5 a^ + 8b^8 a^ + 13b^13 a^ + 21b^21 a^ + 34b
Some examples of Generalized Fibonacci sequences like Catalan’s identity, Cassini’s identity and d’ocagnes’s identity and its properties in [15].
3.2 The Tadpole Graph
A Tadpole graph Tn,k is a graph created by Cn and Pk which is connected by any vertex of the cycle graph Cn that is being pendant of path graph Pk for integers n ≥ 3 and k ≥ 0. For ease of reference we label our corresponding vertices for cycle c 1 ,c 2 ,...,cn and for the path p 1 ,p 2 ,...,pk, such that vertex c 1 and p 1 are adjacent. The Tadpole graph Tn,k read as ”A Tadpole graph with a cycle graph with n vertices and a path graph with k vertices”.
Definition 3.1. The number of independent set in the tadpole graph is denoted by i(Tn,k).
Definition 3.2. The set of all of independent set in the tadpole graph is denoted by I(Tn,k).
Figure 3.5: A Tadpole Graph with a cycle of n vertices and a pendant of k vertices
Main Concepts 12
Example 3.2. T 3 , 2.
Figure 3.6: T 3 , 2
I(T 3 , 2 ) = {∅, {c 1 }, {c 2 }, {c 3 }, {p 1 }, {p 2 }, {c 1 , p 2 }, {c 2 , p 1 }, {c 2 , p 2 }, {c 3 , p 1 }, {c 3 , p 2 }}
Figure 3.7: The set of all independent sets of T 3 , 2
We have i(T 3 , 2 ) = 11.
Example 3.3. T 4 , 1.
Figure 3.8: T 4 , 1
I(T 4 , 1 ) = {∅, {c 1 }, {c 2 }, {c 3 }, {c 4 }, {p 1 }, {c 2 , p 1 }, {c 3 , p 1 }, {c 4 , p 1 }, {c 1 , c 3 }, {c 2 , c 4 }, {c 2 , c 4 , p 1 }}
Figure 3.9: The set of all independent sets of T 4 , 1
We have i(T 4 , 1 ) = 12.
Main Concepts 14
Add an unshaded vertex p 1 followed by shaded vertex p 2.
Figure 3.12: Set of all independent set in T 3 , 2 where p 2 is shaded
I(T 3 , 2 ) = {{p 1 }, {c 1 , p 2 }, {c 2 , p 2 }, {c 3 , p 2 }}. And i(T 3 , 0 ) = 4. For every independent set in I(T 3 , 1 ),
Figure 3.13: The set of all independent sets in T 3 , 1
I(T 3 , 1 ) = {∅, {c 1 }, {c 2 }{c 3 }{p 1 }, {c 2 , p 1 }, {c 3 , p 1 }}. add an unshaded vertex p 2 at the end of the path graph.
Figure 3.14: The set of independent set in T 3 , 2 where p 2 is unshaded
I(T 3 , 2 ) = {∅, {c 1 }, {c 2 }{c 3 }{p 1 }, {c 2 , p 1 }}, {c 3 , p 1 } and i(T 3 , 2 ) = 7.
Therefore i(Tn,k) = i(Tn,k− 1 ) + i(Tn,k− 2 ) = i(T 3 , 1 ) + i(T 3 , 0 ) = 4 + 7 = 11.
We partitioned our path graph into two disjoint subsets such that a path contains the set of shaded vertices and unshaded vertices and we showed that every independent set in I(Tn,k− 1 ) ∩ I(Tn,k− 2 ) is ∅.
Main Concepts 15
Theorem 3.4. i(Tn,k) = i(Tn− 1 ,k) + i(Tn− 2 ,k), for all n ≥ 3 and k ≥ 0
Proof. Now in cycle graph, we need to show that I(Tn,k) = I(Tn− 1 ,k) ∪ I(Tn− 2 ,k) where I(Tn− 1 ,k) ∩ I(Tn− 2 ,k) = ∅. Similar in Theorem 3.3, we partition our tadpole graph Tn,k into two disjoint subsets in cycle graph which contain the set of shaded vertex that is included in independent set and the set of unshaded vertex that is not included in independent set. Label any three consecutive vertices in cycle as { 1 , n − 1 , n} in Tn,k. For every independent set in I(Tn− 2 ,k) which we have two cases:
Case 1: If vertex 1 is shaded, followed by unshaded vertex n − 2, insert a shaded vertex n − 1 followed by unshaded vertex n thus creating independent set in Tn,k that include both vertex 1 and n − 1.
Case 2: Now if vertex 1 is unshaded, insert a shaded vertex n and unshaded vertex n − 1 it creates all independent sets where vertex n is shaded.
For every independent set in I(Tn− 1 ,k) insert an unshaded vertex n which finally creates all independent sets where either vertex 1 and n − 1 is shaded or none of these three vertices 1, n − 1 , n are shaded. Therefore i(Tn,k) = i(Tn− 1 ,k) + i(Tn− 2 ,k)
Main Concepts 17
Case 2. If vertex c 1 is unshaded, add a shaded vertex c 5 in T 3 , 0.
Figure 3.20: T 4 , 0
Creating independent set including vertex c 5
Figure 3.21: The set of all independent set in T 5 , 0 where c 5 is shaded
I(T 5 , 0 ) = {{c 5 }, {c 2 , c 5 }, {c 3 , c 5 }} and i(T 5 , 0 ) = 3. For every independent set of I(T 4 , 0 ),
Figure 3.22: T 4 , 0
I(T 4 , 0 ) = {∅, {c 1 }, {c 2 }, {c 3 }, {c 4 }, {c 1 , c 3 }, {c 2 , c 4 }} and i(T 4 , 0 ) = 7. Add an unshaded vertex c 5 in T 4 , 0.
Figure 3.23: T 5 , 0
Figure 3.24: The set of all independent sets in T 5 , 0 where c 5 is unshaded
Main Concepts 18
I(T 4 , 0 ) ∪ I(T 3 , 0 ) = {∅, {c 1 }, {c 2 }, {c 3 }, {c 4 }, {c 5 }, {c 1 , c 3 }, {c 1 , c 4 }, {c 2 , c 4 }, {c 2 , c 5 }, {c 3 , c 5 }}.
Figure 3.25: The set of all independent sets of I(T 5 , 0 )
Therefore i(Tn− 1 ,k) + i(Tn− 2 ,k) = i(T 4 , 0 ) + i(T 3 , 0 ) = 11.
We partitioned our cycle graph into two disjoint subsets such that a cycle contains the set of shaded vertices and unshaded vertices and we showed that every independent set in I(Tn− 1 ,k) ∩ I(Tn− 2 ,k) is ∅.
Theorem 3.5. i(Tn,k) = Ln+k + Fn− 3 Fk, for all n ≥ 3 and k ≥ 0
Proof. We will show inductively for increasing value in k, For k = 0, then we have a cycle i(Tn, 0 ) = i(Cn) = Ln + Fn− 3 (0) = Ln. For k = 1 we have a cycle and a pendant vertex p 1 at the end of the path graph. i(Tn, 1 ) = i(Cn) + i(Pn− 1 ) = Ln + Fn+1.
Ln + Fn+1 = (Ln+1 − Ln− 1 ) + (Fn + Fn− 1 ) by (3.8) and (3.4) = Ln+1 − (Fn + Fn− 2 ) + Fn− 1 + Fn by (3.11) = Ln+1 − Fn − Fn− 2 + Fn + Fn− 1 = Ln+1 − Fn− 2 + Fn− 1 = Ln+1 + Fn− 3 by (3.6) Ln + Fn+1 = Ln+1 + Fn− 3 F 1
