Further investigation on the super classical mean labeling of graphs obtained from paths

Abstract

Consider the graph G, with the injection Ω from node set to the first p + q natural numbers. Let us assume that the ceiling function of the classical average of the node labels of the end nodes of each link is the induced link assignment Ω^*. If the union of range of Ω of node set and the range of Ω^* of link set is all the first p + q natural numbers, then Ω is called a classical mean labeling. A super classical mean graph is a graph with super classical mean labeling. In this research effort, we attempted to address the super classical meanness of graphs generated by paths and those formed by the union of two graphs.

Keywords

Labeling,super classical mean labeling,super classical mean graph

1 Introduction

A finite, undirected, simple graph G (V, E) containing q edges and p vertices is commonly referred to as a graph during this work. We are using the notations and contexts [10 , 14–16]. We propose [13] for a comprehensive review of graph labeling. P_n represents a path containing n vertices, while C_n represents a cycle containing n vertices. The graph G (V, E) = G₁ ∪ G₂ of graphs G₁ and G₂ with disjoint vertex sets s₁ and s₂ and edge sets E₁ and E₂ is the graph with V = V₁ ∪ V₂ and E = E₁ ∪ E₂ . The graph Tadpole T (n, k) can be identified and generated by finding a vertex of the cycle C_n to an end vertex of the path P_k.

2 Literature survey

Barrientos examined Graceful labelings of chain and corona graphs [9]. Vasuki and Arockiaraj defined the super mean labeling [17] whereas mean labeling [21] of graphs defined by Somosundaram and Ponraj. Durai Baskar et al. proposed the principles of geometric mean labeling [1, 2] of graphs. In [22], Vaidya and Barasara analysed graph harmonic mean labeling. In [5], the authors defined C-exponential mean labeling of various graphs. In [18], Arockiaraj et al. created the idea of F-root square labeling. Vaidya and Prajapati [23] demonstrated the Fibonacci and super graceful labelings of graphs. In [16, 19], the concepts of F-centroidal mean and super F-centroidal labeling [20] were introduced, and their meanness was acknowledged for various standard graphs. Khan et al. delivered the computational and topological properties of neural networks by means of graph-theoretic parameters [3]. Classical mean labeling has been proposed by Muhiuddin et al. [12], whereas it was extended to certain ladder graphs by Alanazi et al. [4] and the authors investigated the classical mean based on duplicating operations [8]. An idea for numerical assessment of specific chemical structures based on the super classical average assignment criterion has been suggested by Rajesh Kannan et al. [7]. Our research focuses on super classical mean labeling methods that utilize graph operation procedures.

3 Methodology

Let an injection Ω : V (G) → {1, 2, 3, …, p + q}. For each edge uv, the induced edge labeling Ω^* is the ceiling function of the average of root square, harmonic, geometric and arithmetic means of the vertex labels of the end vertices of each edge. If Ω (V (G)) ∪ {Ω^* (uv) ; uv ∈ E (G)} = {1, 2, 3, …, p + q}, then Ω is described to as a super classical mean labeling. A super classical mean graph is a graph that employs super classical mean labeling [7]. Here we have demonstrated the super classical meanness by using ceiling function and established the super classical meanness of M (P_n), P_n ∘ S_m, TW (P_n), [P_n ; S₁], arbitrary subdivision of K_1,3, $P_{n}^{2}$ , P_n ∪ C_m and T_n ∪ C_m. Figure 1 displays a classical mean labeling of T (6, 4).

Fig. 1

A super classical mean labeling of T (6, 4).

For T (6, 4), the super classical mean labeling node set and link set are {1, 3, 6, 7, 9, 12, 14, 16, 18} and {2, 4, 5, 8, 10, 11, 13, 15, 17} respectively. Also, the union of the labeled nodes and links set is {1, 2, …, p + q}. Therefore, the super classical mean labeling is admissible on the graph T (6, 4).

4 Main results

The following theorems were proved based on definition of the classical meanness for some special graphs.

Theorem 4.1. The middle graph M (P_n) is a super classical mean graph, for n ≥ 4 .

Proof. Let V (P_n) = {s₁, s₂, s₃, …, s_n} and E (P_n) = {s_αs_α+1 : 1 ≤ α ≤ n - 1}.

p + q = 5n - 5
Values of α			Function	Values
From	To
1	n - 1	Ω (s_α)	5α - 4
1	n - 1	Ω (e_α)	5α - 2
n		Ω (s_α)	5α - 5

Finally, as explained below, the induced edge labeling is accomplished.

Values of α			Function	Values
From	To
1	n - 1	Ω^* (s_αe_α)	5α - 3
1	n - 1	Ω^* (e_αs_α+1)	5α - 1
1	n - 1	Ω^* (e_αe_α+1)	5α

□

Theorem 4.2. The graph P_n ∘ S_m is a super classical mean graph, for n ≥ 1 and m ≤ 3 .

Proof. Let r₁, r₂, …, r_n be the vertices of the path P_n and $v_{1}^{(α)}, v_{2}^{(α)}, \dots, v_{m}^{(α)}$ be the pendant vertices attached at each vertex r_α of the path P_n, for 1 ≤ α ≤ n . Case (i). m = 1.

p + q = 4n - 1
Values of α		Nature of α	Function	Values
From	To
1	n	α ≡ 1 (mod2)	Ω (s_α)	4α - 1
1	n	α ≡ 0 (mod2)	Ω (e_α)	4α - 3
1	n	α ≡ 1 (mod2)	Ω (e_α)	4α - 3
1	n	α ≡ 0 (mod2)	Ω (e_α)	4α - 1

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	n - 1	Ω^* (r_αr_α+1)	4α
1	n - 2	$Ω^{*} (s_{1}^{(α)} r_{α})$	4α - 2

Case (ii). m = 2.

p + q = 6n - 1
Values of α		Function	Values
From	To
1	n	Ω (r_α)	6α - 3
1	n	$Ω (v_{1}^{(α)})$	6α - 5
1	n	$Ω (v_{2}^{(α)})$	6α - 1

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	n - 1	Ω^* (r_αr_α+1)	6α
1	n - 1	$Ω^{*} (s_{1}^{(α)} r_{α})$	6α - 4
1	n - 1	$Ω^{*} (s_{2}^{(α)} r_{α})$	6α - 2

Case (iii). m = 3.

p + q = 8n - 1
Values of α		Function	Values
From	To
2	n	Ω (r_α)	8α - 3
2	n	$Ω (v_{1}^{(α)})$	8α - 8
2	n	$Ω (v_{2}^{(α)})$	8α - 5
2	n	$Ω (v_{3}^{(α)})$	8α - 1
1		Ω (r_α)	3
1		$Ω (v_{1}^{(α)})$	1
1		$Ω (v_{2}^{(α)})$	6

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	n - 1	Ω^* (r_αr_α+1)	8α - 1
1	n	$Ω^{*} (s_{1}^{(α)} r_{α})$	8α - 6
1	n	$Ω^{*} (s_{1}^{(α)} r_{α})$	8α - 4
2	n	$Ω^{*} (s_{3}^{(α)} r_{α})$	8α - 2
1		$Ω^{*} (s_{3}^{(α)} r_{α})$	5

□

Theorem 4.3. The twig graph TW (P_n) is a super classical mean graph, for n ≥ 4 .

Proof. Let s₁, s₂, s₃, …, s_n be the vertices of the path P_n and $r_{1}^{(α)}, r_{2}^{(α)}$ be the pendant vertices at each vertex s_α, for 2 ≤ α ≤ n - 1 .

p + q = 6n - 9
Values of α		Function	Values
From	To
1	2	Ω (s_α)	2α - 1
3	n - 1	Ω (s_α)	6α - 7
3	n - 1	$Ω (u_{1}^{(α)})$	6α - 9
3	n - 2	$Ω (u_{2}^{(α)})$	6α - 5
n - 1		$Ω (u_{2}^{(α)})$	6α - 4
n		Ω (s_α)	6α - 9
2		$Ω (u_{2}^{(α)})$	8
2		$Ω (u_{1}^{(α)})$	6

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	2	Ω^* (s_αs_α+1)	5α - 3
3	n - 2	Ω^* (s_αs_α+1)	6α - 4
2	n - 1	$Ω^{*} (r_{1}^{(α)} s_{α})$	6α - 8
3	n - 1	$Ω^{*} (r_{2}^{(α)} s_{α})$	6α - 6
n		Ω^* (s_α-1s_α)	6α - 11
2		$Ω^{*} (r_{2}^{(α)} s_{α})$	5

□

Theorem 4.4. The graph [P_n ; S₁] is a super classical mean graph, for n ≥ 1 .

Proof. Let r₁, r₂, r₃, …, r_n be the vertices of the path P_n and $s_{1}^{(α)}, s_{2}^{(α)}, s_{3}^{(α)}, \dots, s_{m + 1}^{(α)}$ be the vertices of the star graph S_m such that $s_{1}^{(α)}$ is the central vertex of the star graph S_m, 1 ≤ α ≤ n .

p + q = 6n - 1
Values of α		Function	Values
From	To
1	n	Ω (r_α)	6α - 1
1	n	$Ω (v_{1}^{(α)})$	6α - 3
2	n	$Ω (v_{2}^{(α)})$	6α - 6
1		$Ω (v_{2}^{(α)})$	1

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	n - 1	Ω^* (r_αr_α+1)	6α + 2
1	n	$Ω^{*} (r_{α} s_{1}^{(α)})$	6α - 2
2	n	$Ω^{*} (s_{α} s_{1}^{(α)})$	6α - 5
1		$Ω^{*} (s_{α} s_{1}^{(α)})$	2

□ Theorem 4.5. Arbitrary subdivision of K_1,3 is a super classical mean graph.

Proof. Let G be the graph of arbitrary subdivision of K_1,3.

Case (i). p₁ = p₂, p₁ ≥ 1 and p₃ ≥ 3 .

p + q = 7 + 2p₃ + 2p₂ + 2p₁
Values of β		Function	Values
From	To
2	p₁ + 1	$Ω (v_{1}^{(β)})$	2 (p₁ + p₂) +5 - 4β
1	2	$Ω (s_{β}^{(2)})$	2 (p₁ + p₂) +7 - 4β
3	p₂ + 1	$Ω (v_{2}^{(β)})$	2 (p₁ + p₂) +6 - 4β
1	p₃ + 1	$Ω (v_{3}^{(β)})$	2 (p₁ + p₂) +5 + 2β
0		Ω (s_β)	2 (p₁ + p₂) +5
1		$Ω (s_{β}^{(1)})$	2 (p₁ + p₂)

Finally, as explained below, the induced edge labeling is accomplished.

Values of β		Function	Values
From	To
1	3	$Ω^{*} (s_{0} s_{1}^{(β)})$	2 (p₁ + p₂) +2β
2	p ₁	$Ω^{*} (s_{β}^{(1)} s_{β + 1}^{(1)})$	2 (p₁ + p₂) +3 - 4β
2	p ₂	$Ω^{*} (s_{β}^{(2)} s_{β + 1}^{(2)})$	2 (p₁ + p₂) +4 - 4β
1	p ₃	$Ω^{*} (s_{β}^{(3)} s_{β + 1}^{(3)})$	2 (p₁ + p₂) +6 + 2β
1		$Ω^{*} (s_{j}^{(1)} s_{j + 1}^{(1)})$	2 (p₁ + p₂) -2
1		$Ω^{*} (s_{j}^{(2)} s_{j + 1}^{(2)})$	2 (p₁ + p₂) +1

Case (ii). p₁ < p₂ .

p + q = 7 + 2p₃ + 2p₂ + 2p₁
Values of β		Function	Values
From	To
2	p₁ + 1	$Ω (s_{β}^{(1)})$	2 (p₁ + p₂) +8 - 4β
1	p ₁	$Ω (s_{β}^{(2)})$	2 (p₁ + p₂) +3 - 4β
p₁ + 1	p₂ + 1	$Ω (s_{β}^{(2)})$	2p₂ + 3 -2β
1	p₂ + 1	$Ω (v_{3}^{(β)})$	2 (p₁ + p₂) +5 + 2β
0		Ω (s_β)	2 (p₁ + p₂) +5
1		$Ω (s_{β}^{(1)})$	2 (p₁ + p₃) +3

Finally, as explained below, the induced edge labeling is accomplished.

Values of β		Function	Values
From	To
1	3	$Ω^{*} (s_{0} s_{1}^{(β)})$	2 (p₁ + p₂) +2β
2	p ₁	$Ω^{*} (s_{β}^{(1)} s_{β + 1}^{(1)})$	2 (p₁ + p₂) +6 - 4β
1	p₁ - 1	$Ω^{*} (s_{β}^{(2)} s_{β + 1}^{(2)})$	2 (p₁ + p₂) +1 - 4β
p ₁	p ₂	$Ω^{*} (s_{β}^{(2)} s_{β + 1}^{(2)})$	2p₂ + 2 -2β
1	p ₃	$Ω^{*} (s_{β}^{(3)} s_{β + 1}^{(3)})$	2 (p₁ + p₂) +6 + 2β
1		$Ω^{*} (s_{β}^{(1)} s_{β + 1}^{(1)})$	2 (p₁ + p₂) +5 - 4β

It is determined that the graphs that do not come under Case (i) are super classical mean graphs, and Figure 2, confirms their labeling as the same.

□

Fig. 2

A super classical mean labeling of G with p₁ = p₂ = p₃ = 1, p₁ = p₂ = 1, p₃ = 2 and p₁ = p₂ = p₃ = 2 .

Theorem 4.6. The graph $P_{n}^{2}$ is a super classical mean graph, for n ≥ 3 .

Proof. Let s₁, s₂, …, s_n be the vertices of the path P_n .

p + q = 3n - 3
Values of α		Nature of α	Function	Values
From	To
1	n - 2	α ≡ 1 (mod2)	Ω (s_α)	3α - 2
1	n - 2	α ≡ 0 (mod2)	Ω (s_α)	3α - 3

Values of α		Function	Values
From	To
n		Ω (s_n-1)	3α - 5
n		Ω (s_n-1)	3α - 3

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Nature of α	Function	Values
From	To
1	n - 4	α ≡ 1 (mod2)	Ω^* (s_αs_α+2)	3α + 1
1	n - 4	α ≡ 0 (mod2)	Ω^* (s_αs_α+2)	3α
1	n	α ≡ 1 (mod2)	Ω^* (s_α-3s_α-1)	3α - 9
1	n	α ≡ 0 (mod2)	Ω^* (s_α-3s_α-1)	3α - 8

Values of α		Function	Values
From	To
1	n - 1	Ω^* (s_αs_α+1)	3α - 1
1		Ω^* (s_αs_α-2)	3n - 6

□

Theorem 4.7. The graph G₁ ∪ G₂ is a super classical mean graph where G₁ is a path P_n and G₂ is a cycle C_m, for n ≥ 1 and m ≥ 4 .

Proof. Let r₁, r₂, …, r_m and s₁, s₂, …, s_n be the vertices of the cycle C_m and the path P_n respectively. Take $w = ⌊ \frac{m}{2} ⌋ + 1$ .

Case (i). m ≥ 5 .

p + q = 2n - 1 +2m
Values of α		Function	Values
From	To
1	2	Ω (r_α)	2α - 1
3	$⌊ \frac{m}{2} ⌋$	Ω (r_α)	4α - 6
$⌊ \frac{m}{2} ⌋ + 3$	m	Ω (r_α)	4m + 7 -4α
1	n	Ω (s_α)	2m + 2α - 1

Values of α		Nature of m	Function	Values
From	To
w	m ≡ 1 (mod2)	Ω (r_α)	4α - 6
w	m ≡ 0 (mod2)	Ω (r_α)	4α - 7
w + 1	m ≡ 1 (mod2)	Ω (r_α)	4α - 6
w + 2	m ≡ 0 (mod2)	Ω (r_α)	4α - 8

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	2	Ω^* (r_αr_α+1)	3α - 1
3	$⌊ \frac{m}{2} ⌋$	Ω^* (r_αr_α+1)	4α - 4
1	n - 1	Ω^* (s_αs_α+1)	2m + 2α
m		Ω^* (r₁r_α)	4

Values of α		Nature of m	Function	Values
From	To
w		m ≡ 1 (mod2)	Ω^* (r_αr_α+1)	4α - 4
w		m ≡ 0 (mod2)	Ω^* (r_αr_α+1)	4α - 5
w + 1		m ≡ 1 (mod2)	Ω^* (r_αr_α+1)	4m + 5 -4α
w + 1		m ≡ 0 (mod2)	Ω^* (r_αr_α+1)	4α - 10

Case (ii). m = 4 .

p + q = 2n + 7
Values of α		Function	Values
From	To
1		Ω (r_α)	1
2		Ω (r_α)	2
3		Ω (r_α)	6
4		Ω (r_α)	8

Values of α		Function	Values
From	To
1	n	Ω (s_α)	7 + 2α

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1		Ω^* (r_αr_α+1)	3
2		Ω^* (r_αr_α+1)	5
3		Ω^* (r_αr_α+1)	7
1		Ω^* (r_αr_α+3)	4

Values of α		Function	Values
From	To
1	n - 1	Ω^* (s_αs_α+1)	8 + 2α

□

Theorem 4.8. The graph G₁ ∪ G₂ is a super classical mean graph where G₁ is a graph T_n and G₂ is a cycle C_m, for n ≥ 4 and m ≥ 4 .

Proof. Let r₁, r₂, …, r_m be the vertices of the cycle C_m and s₁, s₂, …, s_n-1 be the vertices of the path P_n-1 and let s_n be the pendant vertex identified with s_n-2 in T_n . Take $w = ⌊ \frac{m}{2} ⌋ + 1$ .

Case(i). m ≥ 5 .

p + q = 2n - 1 +2m
Values of α		Nature of m	Function	Values
From	To
w		m ≡ 1 (mod2)	Ω (r_α)	4α - 6
w		m ≡ 0 (mod2)	Ω (r_α)	4α - 7
w + 1		m ≡ 1 (mod2)	Ω (r_α)	4α - 6
w + 1		m ≡ 0 (mod2)	Ω (r_α)	4α - 8

Values of α		Function	Values
From	To
1	$⌊ \frac{m}{2} ⌋$	Ω (r_α)	2α - 1
$⌊ \frac{m}{2} ⌋ + 3$	m	Ω (r_α)	4m + 7 -4α
1	n - 3	Ω (s_α)	2m + 2α - 1
n - 2		Ω (s_α)	2m + 2α + 1
n - 1		Ω (s_α)	2m + 2α - 4
n		Ω (s_α)	2m + 2α - 1

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1	2	Ω^* (r_αr_α+1)	3α - 1
3	$⌊ \frac{m}{2} ⌋$	Ω^* (r_αr_α+1)	4α - 4
$⌊ \frac{m}{2} ⌋ + 3$	m - 1	Ω^* (r_αr_α+1)	4m + 5 -4α
1	n - 4	Ω^* (s_αs_α+1)	2m + 2α
n - 3		Ω^* (s_αs_α+1)	2m + 2α + 1
n - 2		Ω^* (s_αs_α+1)	2m + 2α
n - 2		Ω^* (s_αs_α+2)	2m + 2α + 2
m		Ω^* (r₁r_α)	4

Values of α		Nature of m	Function	Values
From	To
w		m ≡ 1 (mod2)	Ω^* (r_αr_α+1)	4α - 4
w		m ≡ 0 (mod2)	Ω^* (r_αr_α+1)	4α - 5
w + 1		m ≡ 1 (mod2)	Ω^* (r_αr_α+1)	4m + 5 -4α
w + 1		m ≡ 0 (mod2)	Ω^* (r_αr_α+1)	4α - 10

Case(ii). m=4.

p + q = 2n + 7
Values of α		Function	Values
From	To
1		Ω (r_α)	1
2		Ω (r_α)	3
3		Ω (r_α)	6
4		Ω (r_α)	8
n - 2		Ω (r_α)	2α + 9
n - 1		Ω (r_α)	2α + 4
n		Ω (r_α)	2α + 7

Values of α		Function	Values
From	To
1	n - 3	Ω (s_α)	7 + 2α

Finally, as explained below, the induced edge labeling is accomplished.

Values of α		Function	Values
From	To
1		Ω^* (r_αr_α+1)	2
2		Ω^* (r_αr_α+1)	5
3		Ω^* (r_αr_α+1)	7
1		Ω^* (r_αr_α+3)	4
n - 3		Ω^* (r_αr_α+3)	9 + 2α
n - 2		Ω^* (r_αr_α+3)	8 + 2α
n		Ω^* (r_αr_α+3)	6 + 2α

Values of α		Function	Values
From	To
1	n - 4	Ω^* (s_αs_(α+1))	8 + 2α

□

5 Conclusion

A comprehensive discussion is held regarding the super classical meanness of special graphs derived from paths. For a wide range of other graphs, comparable outcomes can be obtained by using different graph operations. Also, the study of the above labeling on chordal graphs, perfect graphs and line graphs will be interesting.

Authors’ Contributions

A.R.K analyzes the existence of super classical meanness of graphs by collecting the papers, proposing the conjecture, and trying to prove it. G.T edited the manuscript. S.M.K read and approved the final manuscript by verifying it.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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