On the Pythagorean Holes of Certain Graphs

A \emph{primitive hole} of a graph $G$ is a cycle of length 3 in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of the graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of Pythagorean holes of graphs and initiate some interesting results on Pythagorean holes in general as well as results in respect of set-graphs and Jaco graphs.


Introduction
For general notations and concepts in graph theory, we refer to [1], [4] and [9]. All graphs mentioned in this paper are simple, connected undirected and finite, unless mentioned otherwise.
A hole of a simple connected graph G is a chordless cycle C n , where n ∈ N , in G. The girth of a simple connected graph G, denoted by g(G), is the order of the smallest cycle in G. The following notions are introduced in [5].
Definition 1.1. [5] A primitive hole of a graph G is a cycle of length 3 in G. The number of primitive holes in a given graph G is called the primitive hole number of that graph G. The primitive hole number of a graph G is denoted by h(G). Definition 1.2. [5] The primitive degree of a vertex v of a given graph G is the number of primitive holes incident on the vertex v and the primitive degree of the vertex v in the graph G is denoted by d p G (v).
Some studies on primitive holes of certain graphs have been made in [5]. The number of primitive holes in certain standard graph classes, their line graphs and total graphs were determined in this study. Some of the major results proved in [5] are the following. Definition 1.5. [6] Let A (n) = {a 1 , a 2 , a 3 , . . . , a n }, n ∈ N be a non-empty set and the i-th s-element subset of A (n) be denoted by A It can be noted from the definition of set-graphs that A (n) = ∅ and if |A (n) | is a singleton, then G A (n) to be the trivial graph. Hence, all sets we consider here are non-empty, non-singleton sets.
Let us now write the vertex set of a set-graph G A (n) as V (G A (n) ) = {v rs : 1 ≤ r ≤ n s }, where s is the cardinality of the set corresponding to the vertex v rs . The r-th s-element subset of A (n) , which corresponds to the vertex v rs is denoted by A It is proved in [6] that any set-graph G has odd number of vertices. Some other important properties and results established in [6] are as follows. An important property on set-graphs is that the the vertices of a set-graph G, corresponding to the sets of equal cardinality, have the same degree. Also, the vertices in a set-graph G, corresponding to the singleton subsets subsets of A (n) , are pairwise non-adjacent in G.
For any vertex v of a set-graph G = G A (n) , it has been proved that 2 n−1 − 1 ≤ d G (v) ≤ 2 n − 2 or equivalently, we have ∆(G) = 2 δ(G). It has also been proved that there exists a unique vertex v in a set-graph G A (n) having the highest possible degree. It can also be noted that the maximal degree of vertex in a set-graph G is always an an even number and the minimal degree of a vertex in G is always an odd number.
Another important result proved in [6] is that the set-graph G A (n) , n ≥ 2 has exactly two largest complete graph, K 2 n−1 .
Some major results on primitive hole number and primitive degree of set-graphs (see [6]) are the following.
The primitive degree of a vertex v of a set-graph The following theorem on tightness number has been established in [6]. Theorem 1.6. Consider the set-graph G A (n) , n ≥ 1 and extend to the set-graph As an extension to the studies made in [5] and [6], in this paper, we propose a new parameter called the Pythagorean holes of a graph. Further to some general results, we also discuss this parameter in respect of set-graphs and Jaco graphs.

Pythagorean Holes of Graphs
By a Pythagorean triple of positive integers, we mean an ordered triple (a, b, c), where a < b < c, such that a 2 + b 2 = c 2 . Also, if (a, b, c) is a Pythagorean triple of positive integers, then for any (positive) integer k, the triple (ka, kb, kc) is also a Pythagorean triple. That is, we have (ka) 2 + (kb) 2 = (kc) 2 .
Using the concepts of Pythagorean triples, we now introduce the notion of Pythagorean holes of a given graph G as follows.
Let us denote the number of Pythagorean holes of a graph G by h p (G).
We can easily construct a graph with a Pythagorean hole as follows. Let (n 1 , n 2 , n 3 ) be a Pythagorean triple. Draw a triangle, say C 3 , on the vertices v 1 , v 2 , v 3 . We extend this triangle to a graph G where d G (v i ) = n i ; 1 ≤ i ≤ 3 as follows. Attach n 1 − 2 pendant vertex to the vertex v 1 , attach n 2 − 2 pendant vertices to the vertex v 2 and add n 3 − 2 pendant vertices to v 3 to obtain a new graph graph G. Here, G is a unicyclic graph on n 1 + n 2 + n 3 − 3 vertices and edges each and has one primitive hole. Therefore, the triangle v 1 v 2 v 3 v 1 is a Pythagorean hole in G.
By a minimal graph with respect to a given property, we mean a graph with minimum order and size satisfying that property. In view of this concept we introduce the following notion.
Definition 2.2. A graphical embodiment of a given Pythagorean triple is the minimal graph that consists of a Pythagorean hole with respect to that Pythagorean triple.
Clearly, the graph G mentioned above is not the graphical embodiment of the Pythagorean triple (n 1 , n 2 , n 3 ). Verifying the existence of a graphical embodiment to a given Pythagorean triple is an interesting question that leads to the following theorem. Proof. Let (n 1 , n 2 , n 3 ) be a Pythagorean triple of positive integers such that n 1 < n 2 < n 3 . First, draw a triangle on vertices v 1 , v 2 , v 3 . Now, plot n 1 − 2 vertices and attach them to the vertex v 1 so that d(v 1 ) = n 1 . Now, attach the n 1 − 2 vertices to v 2 and v 3 also. At this step, d(v 2 ) = d(v 3 ) = n 1 . Now, the n 2 − n 1 additional edges are required to be incident on the vertex v 2 . Hence, plot new n 2 − n 1 vertices and attach them to v 2 and v 3 . Now, d(v 2 ) = n 2 , as required. But, here d(v 3 ) = n 2 and additionally n 3 −n 2 edges are to be incident on v 3 . Hence, create new Clearly, this graph is the smallest graph with a Pythagorean hole corresponding to the given Pythagorean triple. Any graph other than G will have more vertices than G. Hence, G is a unique graphical embodiment of the given Pythagorean triple.
The graph G in Figure 1 is an example for a graph containing a Pythagorean hole corresponding to a Pythagorean triple (3,4,5). The graph G has the minimum number of vertices (that is, 6 vertices) required to contain a Pythagorean hole.

Figure 1
The uniqueness of the Pythagorean hole in a graphical embodiment of a given Pythagorean triple is established in the following proposition. Proof. Let (n 1 , n 2 , n 3 ) be a Pythagorean triple of positive integers and let G be a graphical embodiment of this triple obtained as explained in Theorem 2.
It is to be noted that all new vertices that are adjacent to v 1 in G will be adjacent to both v 2 and v 3 also and hence are of degree 3. Similarly, all new vertices that are adjacent to v 2 is adjacent to v 3 also. Therefore, the degree of the vertices that are adjacent to v 2 , but not to v 1 , is 2 and the degree of the vertices that are adjacent only to v 3 is 1. Also, no two of these new vertices are mutually adjacent. Hence, for any three vertices The characteristics of the graphical embodiment of a Pythagorean triple seems to be much promising in this context. The size and order of the graphical embodiment are determined in the following result.
Theorem 2.5. Let G be the graphical embodiment of a given Pythagorean triple (n 1 , n 2 , n 3 ), with usual notations. Then, (i) the order (the number of vertices) of G is one greater than the highest number in the corresponding Pythagorean triple.
(ii) the size (the number of edges) of G is three less than the sum of numbers in the corresponding Pythagorean triple.
Proof. Let G be a graphical embodiment of a Pythagorean triple (n 1 , n 2 , n 3 ). Let v 1 , v 2 , v 3 be the vertices with d(v 1 ) = n 1 , d(v 2 ) = n 2 , and d(v 3 ) = n 3 . As explained in Theorem 2.3, n 1 −2 vertices are attached to v 1 , v 2 , and v 3 , further n 2 −n 1 vertices are attached to v 2 and v 3 and n 3 − n 2 pendant vertices are attached to v + 3. Then, That is, the order of the graphical embodiment is one greater than the highest number in the corresponding Pythagorean triple.
(ii) Let V 1 be the set of all newly introduced vertices which are adjacent to all three vertices v 1 , v 2 , and v 3 . Therefore, for all vertices x in V 1 , we have d(x) = 3. Therefore, . Similarly, let V 2 be the set of new vertices which are adjacent to v 2 , and v 3 . For all vertices y in V 2 , we have d(y) = 3. Therefore, y∈V 2 d(y) = 2(n 2 − n 1 ). Also, let V 3 be the set of new vertices that are adjacent to v 3 only. Here, for all vertex z in V 3 , we have d(z) = 1 and hence Therefore, v∈V (G) d(v) = n 1 + n 2 + n 3 + 3(n 1 − 2) + 2(n 2 − n 1 ) + (n 3 − n 2 ) = 2(n 1 + n 2 + n 3 − 3). Since for any connected graph G, we have |E(G)| = n 1 + n 2 + n 3 − 3. That is, the size of G is three less than the sum of numbers in the Pythagorean triple.
This completes the proof.
Theorem 2.6. The primitive hole number of the graphical embodiment G of a Pythagorean triple is h(G) = 2n 1 + n 2 − 5.
Proof. Let G be a graphical embodiment of a Pythagorean triple (n 1 , n 2 , Then, every vertex in V 1 forms a triangle with any two vertices among v 1 , v 2 and v 3 . Hence each vertex in V 1 corresponds to three triangles in G. Therefore, the total number of such triangles is 3(n 1 − 2). Similarly, every vertex in V 2 , being adjacent only to v 2 and v 3 , forms a triangle in G and no vertex in V 3 is a part of a triangle in G, Therefore, the total number of triangles in G is h(G) = 1+3(n 1 −2)+(n 2 −n 1 ) = 2n 1 + n 2 − 5. This completes the proof.
The following theorem discusses certain parameters of the graphical embodiments of the given Pythagorean triples. (ii) the independence number of G is two less than the highest number in the Pythagorean triple. That is, α(G) = n 3 − 2.
(iii) the covering number of G is β(G) = 3.
(iv) Clearly, the vertex v 3 is adjacent to all other vertices in the graphical embodiment G, we have γ(G) = 1.
This completes the proof.
It is noted from Theorem 2.7, the chromatic number, covering number and domination number of the graphical embodiment of any Pythagorean triple are always the same.
An immediate consequence of the general Pythagorean property is that the vertices of Pythagorean holes can be mapped onto interesting Euclidean geometric objects which we can construct along the sides of a right angled triangle or along the surfaces of the corresponding right angled prism. where (a, b, c) is a Pythagorean triple of positive integers. Then, Since 1 8 π is a constant, we have That is, the Pythagorean property holds for the triplet (d G (v 1 ), d G (v 2 ), d G (v 3 )). Geometrically it means that the area of the semi-circle with length of hypotenuse as its diameter is equal to the sum of the respective areas of the semi-circles with lengths of the other two sides of a right angled triangle as the diameters. where (a, b, c) is a Pythagorean triple of positive integers. Then,

Illustration 2. For an arbitrary real number
Since 1 2 a 2 + abc + 1 2 b 2 + abc = 1 2 c 2 + 2abc, it implies geometrically that the area under the straight line f (x) = x + 2ab, x ∈ R between the limits x = 0 and x = c is equal to the sum of the respective areas under the straight lines f (x) = x+bc, x ∈ R between the limits x = 0 and x = a and f (x) = x + ac, x ∈ R between the limits x = 0 and x = b in respect of a right angled triangle.
Clearly, we can find many such mappings and it would be worthy to find the applications of these mappings.

On Pythagorean Holes of Set-Graphs
The following result is on the degree sequence of set-graphs.

Theorem 3.2. A set-graph has no Pythagorean holes.
Proof. Note that G A (1) and G A (2) have no holes and hence no Pythagorean holes. Now, consider the set-graph G A (3) which has degree sequence (3,3,3,5,5,5,6). Ev- 3) has no Pythagorean hole. Now, assume that the result is true for G A (m) , m being an arbitrary positive integer. We have to verify whether the theorem is true for n = m + 1. For this, extend G A (m) to G A (m+1) . Invoking Theorem 1.6, we need to prove the result in respect of the Pythagorean triple (3,4,5) or any Pythagorean triple of the form (3l, 4l, 5l); l ∈ N. Here we have to consider the following cases.
Case 1: Consider any triplet (d i , d j , d k ) from the degree sequence of the erstwhile G A (m) . Here, we only consider triples (d i , d j , d k ) for which 7 ≤ d i < d j < d k .
Since 2d i + 1 = 3d i holds if and only if d i = 1 < 7, we do not have a natural number (positive integer) sufficiently large. Also, 2d j + 1 = 4d j if and only if d j = 1 2 / ∈ N. Finally, 2d k + 1 = 5d k if and only if d k = 1 3 / ∈ N. Hence, the vertices with respective degrees corresponding to the triplet (d i , d j , d k ) in erstwhile G A (m) , do not represent a Pythagorean hole in G A (m+1) . Subcase 1.1 : Consider the triplet (2d i + 1, 2d j + 1, 2 m + d k ) for some corresponding replica vertex. The first two entries disqualify the triplet to represent a Pythagorean hole in G A (m+1) . Subcase 1.2 : Consider the triplet (2d i + 1, 2 m + d j , 2 m + d k ) for some corresponding replica vertices. The proof follows similar to that of Subcase 1.
We denote a finite Jaco graph by J n (1) and its underlying graph by J * n (1). In both instances we will refer to a Jaco graph and distinguish the context by the notation J n (1) or J * n (1). The primitive hole number of the underlying graph of a Jaco graph has been determined in the following theorem (see [5]).
[5] Let J * n (1) be the underlying graph of a finite Jaco Graph J n (1) with Jaconian vertex v i , where n is a positive integer greater than or equal to 4.
It can be noted that the smallest Jaco graph having a Pythagorean hole is J * 8 (1). Theorem 4.3. For any primitive hole of the Jaco graph J * n (1), n ∈ N on the vertices v i , v j , v k with i < j < k we have a primitive hole on the vertices v li , v lj , v lk in J * n≥lk (1), l ∈ N. Proof. For any primitive hole of the Jaco graph J * n (1), n ∈ N on the vertices The latter implies the edge v li v lk exists in the Jaco graph J * n≥lk (1), l ∈ N. Because the subgraph induced by vertices From the definition of Jaco graphs, it easily follows that the Pythagorean triples labelled t i ; 1 ≤ i ≤ 5 are applicable to the Pythagorean holes found in Jaco graphs. We shall refer to these Pythagorean triples as type i ; i ∈ N. With regards to Theorem 4.2, we shall refer to lt i = (a i l, b i l, c i l) as type i as well. The number of Pythagorean holes of type i in a graph will be denoted h p t i (G). Other Pythagorean triples generated by Euclid formula which are not of a specific primitive type offer some additional Pythagorean holes in Jaco graphs. Let us denote these additional types as e i ; i ∈ N.

Conclusion and Scope for Further Studies
We have discussed particular types of holes called Pythagorean holes of given graphs studied the existence thereof in certain graphs, particularly in set-graphs and Jaco graphs.
As all graphs do not contain Pythagorean holes, the characteristics and structure of graphs containing Pythagorean holes arouses much interest and extends a lot further studies. The questions regarding the number of Pythagorean holes in a given graph, if exists, is a parameter that needs to be studied further.
We proved that the graphical embodiment of Pythagorean triples contain exactly one Pythagorean hole. But, some graphs may contain more than one Pythagorean hole. The study on the graphs or graph classes containing more than one Pythagorean classes corresponding to one or more Pythagorean triples demands further investigation.
The study seems to be promising as it can be extended to certain standard graph classes and certain graphs that are associated with the given graphs. More problems in this area are still open and hence there is a wide scope for further studies.