Chromatic Zagreb indices for graphical embodiment of colour clusters

For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, where $\mathcal{C}_i$ is a colour class such that $|\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\chi(G^{\mathbb{C}}_1)=\chi(G^{\mathbb{C}}_2)=\ell$ and $\min\{\varepsilon(G^{\mathbb{C}}_1)\}=\min\{\varepsilon(G^{\mathbb{C}}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$. Therefore, the problem is the edge-minimality inverse to finding the chromatic number of a given simple connected graph. In this paper, we also discuss the chromatic Zagreb indices corresponding to $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$.


Introduction
For general notation and concepts in graphs and digraphs see [3,4,6,12]. Unless mentioned otherwise all graphs we consider in this paper are finite, simple, connected and undirected graphs.
The chromatic number χ(G) ≥ 1 of a graph G is the minimum number of distinct colours that allow a proper colouring of G. Such a colouring is called a chromatic colouring. of G. The colour weight of a colour in a graph G is the number of times a colour c j has been allocated to the vertices of G, denoted by θ(c j ) ≥ 1. In other words, the colour weight of a colour c i can be considered as the cardinality of the corresponding colour class in the graph concerned. Analogous to set theory notation, let C = 1≤i≤χ(G) C i , and be called a colour cluster. It is noted that for χ(G) ≥ 2 a graph G C with max{ε(G C )} that requires the colour cluster C to allow exactly a chromatic colouring (minimum proper colouring) is the complete χ(G)-partite graph K θ(c 1 ),θ(c 2 ),...,θ(c χ(G) ) or put differently, the complete -partite graph with vertex partitioning V i (G C ), 1 ≤ i ≤ such that |V i (G C )| = |C i |, and is a complete graph, K χ(G) . For the colour cluster C = (C 1 ) the graph G C is a null graph (no edges). Energy can entirely be transformed and transmitted as energy waves. The electromagnetic spectrum of these energy waves ranges between cosmic rays, gamma rays, X-rays, ultra violet rays, optical light with its inherent spectrum of optical colours, infra-red rays, micro waves, short radio waves and long radio waves. These waves or rays transmit at the speed of light. An energy unit per wave (or ray) type is the amount of energy transmitting per second. All such energy units are simply called, colours. If for a given finite cluster of distinct energy units (colours), the colours must be linked to ensure connectivity but same colours must remain apart to prevent merging, then the problem we investigate is that of finding a simple connected graphical embodiment allowing the colours as a proper colouring. More specifically the problem to be considered is to find properties and related results of a connected graph G C with min{ε(G C )} that allows the prescribed colour cluster C as a chromatic colouring. Henceforth the graph G C will mean a graph with minimum edges allowing C as a chromatic colouring.

Graphical Embodiment of C
We note that the cycle C 5 has χ(C 5 ) = 3 with colour cluster say, C = (c 1 , c 1 , c 2 , c 2 , c 3 ). However, path P 5 is a graph with minimum number of edges that allows C as a proper colouring (not chromatic colouring). This observation leads us to our first useful result given below.
Lemma 2.1. For any colour cluster C = (C 1 , C 2 , C 3 , . . . , C ) with |C i | = r i > 0, ≥ 2, 1 ≤ i ≤ , the connected graphical embodiment with minimum edges that allows C a proper colouring has . Add the edges v 1,1 v j,k , for all 2 ≤ j ≤ , 1 ≤ k ≤ r j . Also, add the edges v 1,i v 2,1 , 2 ≤ i ≤ r 1 . Clearly, the graph G 1 obtained is a tree for which C is a proper colouring. Also, since G 1 is a connected acyclic graph, it has minimum number of The construction methodology mentioned in the proof of Lemma 2.1 may be called the Type-I graphical embodiment. In order to obtain a G C , we consider the induced star subgraph v 1,1 , v 2,1 , v 3,1 , . . . , v ,1 of G 1 and add the edges to obtain a complete subgraph, K . Note that 1 2 ( − 1)( − 2) edges were added to the graph G 1 .
An alternative construction of such a graph G C can be done by utilising the initial edges v i,1 v i+1,j , 1 ≤ i ≤ − 1, 1 ≤ j ≤ r i+1 as well as the edges v 2,1 v 1,j , 2 ≤ j ≤ r 1 to obtain graph G 2 . The aforesaid construction is called the Type-II graphical embodiment. Thereafter, consider the induced path subgraph v 1,1 , v 2,1 , v 3,1 , . . . , v ,1 and add the required 1 2 ( − 1)( − 2) edges to obtain the complete subgraph, K . This brings us to the existence of the following theorem.
Theorem 2.2. For any colour cluster C, at least one graphical embodiment exist with minimum number of edges for which C is a chromatic colouring and χ(G C ) = .
Proof. As stated above, for the colour cluster C = (C i ) 1 ≤ i ≤ and |C i | = |C j | = 1, for all values of i, j, the graph G C is complete. Hence, at least one graph with required properties exists.
For all other colour clusters, the result is a direct consequence of Lemma 2.1 and the alternative construction of G C .
We note that for a colour cluster C = (C 1 , C 2 ) the graph G C is acyclic. For colour clusters C = (C i ), i ≥ 3 the graphical embodiment G C has a unique maximum clique. Figure 1(a) and Figure 1   imb(e), where imb(e) = |d(v) − d(u)|. In [5], this irregularity index was named the third Zagreb index to conform with the terminology of chemical graph theory. Recently, the topological index called total irregularity has been introduced in [1] as irr t (G) = 1 This irregularity index may be called the fourth Zagreb index. The study of Zagreb indices is strongly dependent on the structural property of the edges and subsequently the degree of the vertices of the graph under study. A vertex colouring with colours of minimum subscripts is called a minimum parameter colouring. Unless stated otherwise, the colour sets we consider in this paper are minimum parameter colour sets. The weight of a colour c i is defined to be the number of times the colour c i is allocated to vertices of a graph G and is denoted by The notion of chromatic Zagreb indices was introduced in [9]. For ease of reference we also recall that the first three Zagreb indices are defined as: We define the default values, M 1 (K 1 ) = M 2 (K 1 ) = M 3 (K 1 ) = 0. Note that the Zagreb indices are all functions of the degree of vertices of graph G. For a given graph, the vertex degrees are invariants.
Note that for any minimum parameter set of colours C, |C| = , a graph G has ! minimum parameter colourings. Denote these colourings ϕ t (G), 1 ≤ t ≤ !. Define the variable chromatic Zagreb indices as: From the above we define the minimum and maximum chromatic Zagreb indices as follows: In view of the above mentioned terminology, we also observe the following useful lemma.
Proof. Because the M ϕt 1 (G) is defined in terms of the colour subscript only and χ(G) = χ(G + e) the result follows immediately.

Application to G C
In this discussion, we will consider a specific initiation colour cluster from amongst the ! orderings of the colour classes. That is, we consider a colour cluster C = r j → r f (i) . Hence, the notation styles may be used interchangeably depending on which notation proffers the idea best. The convention is to determine the minimum chromatic Zagreb index directly in terms of the initiation colour cluster and the definition of construction. Determining the maximum chromatic Zagreb index will follow from an appropriate colour mapping. First, we recall important results that have been proved in [9].
For a finite tree of order n ≥ 4, we have Applying Lemma 3.1 to the G C , where C = (C 1 , C 2 ), we have the next result as follows.
, the result follows through substitution and simplification.
(ii) Since n 2 Note that both Type-I and II construction of a graph G with minimum edges which allows C as a proper colouring have χ(G) ≤ . This will be true for any other construction type. Therefore, each construction type is extended to a well defined complete subgraph to obtain G C , except for the case = 2. From [9] the chromatic Zagreb indices for complete graphs are known.
Theorem 3.6. Any two graphs G C 1 , G C 2 which allow the colour cluster C as a chromatic colouring have Proof. The result is a direct consequence of the definition.
The following theorem discusses whether a thorn complete graph can be the graphical embodiment of a colour cluster. Proof. Since, χ(K ) = and a minimum number of pendant vertices exist to allocate exactly |C i | − 1, 1 ≤ i ≤ colours to pendant vertices, the result follows immediately.
Note that Theorem 3.7 provides a sufficient condition only. We know that all odd cycles C n , n ≥ 5, have chromatic number 3. These cycles are triangle-free and have min{ε(C n )} for the corresponding colour cluster ((c 1 , c 1 where n = 2t + 1.
We extend this study by analysing the chromatic Zagreb indices for the Type-I and II constructions for a colour cluster C = (C 1 , 3.2 Analysis of chromatic Zagreb indices for G C 1 Proposition 3.8. For the colour cluster C the graph G 1 obtained from the Type-I construction has Proof. (i) Follows directly from the definition of the first chromatic Zagreb index.
(ii) Each edge v 2,1 v i,j , 2 ≤ j ≤ r 1 has product value of 2. Therefore the first term. The second term follows by similar reasoning in respect of each colour c i , 2 ≤ i ≤ . Therefore the second term hence, the result for the minimum second chromatic Zagreb index follows. The maximum second chromatic Zagreb index follows similarly in respect of the map c i → c −(i−1) .
(iii) The minimum third chromatic Zagreb index follows by similar reasoning as in part (ii) and the maximum third chromatic Zagreb index follows in respect of the map c → c 1 , c 1 → c 2 , c 2 → c 3 . . . c −2 → c −1 .
Corollary 3.9. For the colour cluster C, the graph G C 1 has: Proof. For parts (i), (ii) and (iii), the additional sums terms follow from the known results for the corresponding complete graph on vertices v j,1 , 2 ≤ j ≤ − 1.

2
Proposition 3.10. For the colour cluster C the graph G 2 obtained from the Type-II construction has Proof. (i) Follows directly from the definition of the first chromatic Zagreb index.
(ii) The first term follows from the edges v 2,1 v 1,i , 2 ≤ i ≤ r 1 . The second term follows directly from the definition of the second chromatic Zagreb index.
(iii) Define the map c 1 → c , c 2 → c 1 , c 3 → c −1 , c 4 → c 2 , c 5 → c −2 and so on. Clearly, this mapping ensures maximal adjacency differences for all combinations of the distinct colours. Therefore the result follows through similar reasoning used in part (ii).
Corollary 3.11. For the colour cluster C, the graph G C 2 has Proof. For parts (i), (ii) and (iii), the additional sums terms follow from the additional edges added, on completing the graph on the path v 1,i , 1 ≤ i ≤ . Note |i − j|) which then simplifies to the result.

Application to Certain Positive Non-Increasing Finite Integer Sequences
In this section, we consider two positive non-increasing finite integer sequences. It lays the basis for investigating other such sequences.

Positive integers
The first sequence is defined as a mirror image of the positive integers. Let s 1 = {a n }, a i = − (i − 1), 1 ≤ i ≤ . Also, let θ(c i ) = a i . Consider the corresponding colouring cluster C = (C i ), 1 ≤ i ≤ .
Theorem 4.1. Let C = (C i ), 1 ≤ i ≤ , and θ(c i ) = a i , a i ∈ s 1 . Then, for the colour cluster C the graph G C 1 obtained from the Type-I construction has and for the colour cluster C the graph G C 2 obtained from the Type-II construction has find partial closure of the results above. For ex- . The partial closure of the other results can also be established in a straight forward manner.

Fibonacci numbers
The second sequence we consider is the mirror image of the Fibonacci sequence.
Theorem 4.2. Let C = (C i ), 1 ≤ i ≤ , and θ(c i ) = a i , a i ∈ s 2 . Then, for the colour cluster C the graph G C 1 obtained from the Type-I construction has and for the colour cluster C the graph G C 2 obtained from the Type-II construction has f 4 i = f 4 +2 + 4(−1) +1 · f 2 +1 + 6 +3 find partial closure of the results above. These exercises are left for the reader. For useful references see [7,10,13].

Conclusion
Note that any tree T that allows C as a proper colouring is a construction type, say Type-i. It then follows that any connected subtree T ⊆ T of order that has exactly one coloured vertex of each distinct colour in C, can be completed by adding exactly 1 2 ( − 1)( − 2) edges to obtain G C . It is important to note that not all constructions types have this property and therefore, cannot yield the graphical embodiment G C .  c 4 )). The path v 1,1 v 2,1 v 1,2 v 2,2 v 1,3 v 2,3 v 1,4 v 2,4 v 1,5 v 3,1 v 1,6 v 3,2 v 4,1 v 3,3 v 4,2 represents a path-type construction to obtain G p with minimum edges allowing C as a proper colouring. However, a path subgraph that contains colours c 1 , c 2 , c 3 cannot also contain colour c 4 . Similarly, a path subgraph that contains colours c 1 , c 3 , c 4 cannot also contain colour c 2 . Finally, path subgraphs that contain either c 2 , c 3 , c 4 or c 1 , c 2 , c 4 do not exist. Therefore, we need more than 1 2 ( − 1)( − 2) additional edges in order to construct G C p .
The definitions of minimum and maximum chromatic Zagreb indices are reliant on the premises of a given (fixed) graph construction. This study generalised to two primary variants i.e. graph construction and colour cluster for a given number of distinct colours. Note that the respective |C i | = r i = θ(c i ) ≥ 1, ∀ i ∈ N.
Hence, although this study considered the Type-I and II construction followed by the analysis of chromatic Zagreb indices of both the positive integers and the Fibonacci sequence, the construction type which results in the min{min}, max{min} and the min{max}, max{max} values for the chromatic Zagreb indices M C − i , M C + i , i = 1, 2, 3 respectively, remains elusive at this stage. Generally, the results are computationally complex and it is indicative that deeper results will most likely require advanced computer analysis. The application to colour clusters for which the colour weights represent other defined non-negative integer sequences or specialised number sequences opens a wide scope for further research.
Finally it is suggested that the Type-II graphical embodiment can be described recursively as the colour cluster extends from C = (C i ), 1 ≤ i ≤ to C = (C i ), 1 ≤ i ≤ + 1.