Perfectly antimagic total graphs

Abstract

An one-one correspondence function λ from V(G) ∪ E(G) to the set {1, 2, …, |V(G) | + |E(G) |} is a total labeling of a finite undirected graph G without loops and multiple edges, where |V(G) |and |E(G) | are the cardinality of vertex and edge set of G respectively. A perfectly antimagic total labeling is a totally antimagic total labeling whose vertex and edge-weights that are also pairwise distint. Perfectly antimagic total (PAT) graph is a graph having such labeling. The topic of discovering perfectly antimagic total labeling of some families of graphs is discussed in this paper. We also came up with certain conclusions about dual of a perfectly antimagic total graphs. Finally, we provided that the necessary and sufficient condition for a dual of a regular and irregular PAT graph to be a PAT graph.

Keywords

Totally antimagic total labeling antimagic total labeling total labeling

1 Introduction

All graphs throughout this work are assumed to be finite with no loops, no multiple edges and no directions. Labeling of graph is a function that converts a collection of graph elements into a set of positive integers. If a function’s domain is the edge or vertex set of a (p, q)-graph G, the function is referred to be a edge or vertex labeling of G, where p and q are the number of vertices and edges of G respectively. If an one-one correspondence function’s domain is the union of edge and vertex set of G, then the labeling is called total labeling of G. The vertex and edge weight of a vertex v and an edge uv under total labeling is defined by ${wt}_{λ} (v) = λ (v) + \sum_{w \in N (v)} λ (vw) and$ ${wt}_{λ} (uv) = λ (u) + λ (v) + λ (uv)$ respectively, where N(v) is the neighbour set of v. If weights of all edges of G are pairwise distinct under total labeling λ, then λ is called an edge antimagic total labeling (EAT) of G. Similarly if weights of all vertices are pairwise distinct under total labeling λ, then λ is called a vertex antimagic total labeling (VAT) of G.

The concept of an antimagic labeling of a graph is introduced in [6]. An edge labeling of a graph G is an vertex antimagic edge (VAE) labeling in which all of the vertex-weights must be pairwise distinct. The total of the labels of all edges that intersect the vertex is the vertex-weight for edge labeling. Every cycle is VAE labeling, disjoint union of two VAE graph is VAE graph and every simple 2-regular graph is VAE graph was studied in [4]. The concept of an VAT labeling was introduced by Baca, Bertault, MacDougall, Miller, Simanjuntak and Siamin [2]. Super VAT labelings for disconnected graphs were shown in [1] respectively. All graphs have vertex antimagic total labelings was proved in [8]. For more information on graph labelings, see [5].

In 2015, M. Miller, O. Phanalasy, J. Ryan, Andrea Semanicova-Fenovcikova and Anita Abildgaord Sillasen [3] introduced the concept of totally antimagic total (TAT) graph. If a total labeling is both edge and vertex antimagic then the labeling is called TAT labeling. A graph with TAT labeling is called TAT graph. We can question if there is a TAT labeling where the vertex and edge-weights that are also pairwise distinct. Then such labeling will be referred to as perfectly antimagic total labeling, and a graph with such labeling will be referred to as perfectly antimagic total (PAT) graph. The PAT graph is an extension of the TAT graph concept. Moreover, every PAT graph is a TAT graph, but not every TAT graph is a PAT graph. In recent days, a lot of the most harmful viruses have affected people with various symptoms. Let the vertices be such viruses and the edges be their symptoms. The label of the edge indicates the number of people who share a common symptom with any two viruses, while the label of the vertex indicates the number of people affected by a specific virus only. The vertex weight predicts how many people have a certain virus and all of its associated symptoms, whereas the edge weight predicts how many people have a specific symptom and are infected by both of the viruses that cause it. The PAT concept is used to calculate the precise number of people affected by each virus and each symptom in various aspects.

In order to remedy this real-life situation, we need to focus more and less on the viruses and symptoms that have affected the maximum and minimum number of people, respectively. Using the PAT concept, it is easy to find the maximum and minimum number of people affected by various viruses and their symptoms in various aspects. In the future, the PAT concept will be implemented as software. Using that software, we can easily calculate the above details in a short period of time. In addition to the healthcare sector, PAT will also be used in biological, crystallographic, and other disciplines in the future. It is also used to improve the mechanism for scheduling international assignments. In the future, we’ll introduce a few new PAT parameters that are crucial for optimization techniques.

2 PAT labeling of some families of graphs

The existence of PAT labeling of some families of graphs are mentioned in this section.

Theorem 2.1. The path graph P_n is PAT, for n ≥ 2. Proof. Let v₁, v₂, …, v_n be the vertices of P_n, for n ≥ 2 and let λ be the total labeling of P_n with $\begin{matrix} λ (v_{i}) & = & i, for i = 1, 2, \dots, n \\ λ (v_{i} v_{i + 1}) & = & 2 n - i, for i = 1, 2, \dots, n - 1 \end{matrix}$ For 2 ≤ i ≤ n - 1, we get $\begin{matrix} {wt}_{λ} (v_{1}) & = & 2 n, \\ {wt}_{λ} (v_{n}) & = & 2 n + 1, \\ {wt}_{λ} (v_{i}) & = & 4 n + 1 - i \end{matrix}$ We can conclude from the preceding that ${wt}_{λ} (v_{1}) < {wt}_{λ} (v_{n}) < {wt}_{λ} (v_{i}) .$ i.e, vertex-weights are pairwise distinct. For 1 ≤ i ≤ n - 1, we get wt_λ(v_iv_i+1) =2n + 1 + i. i.e, edge-weights are pairwise distinct, since edge-weights are dependent on i and i varies from 1 to n - 1. For 1 ≤ i ≤ n - 1, we obtain wt_λ(v_iv_i+1) > wt_λ(v₁) and wt_λ(v_iv_i+1) > wt_λ(v_n). For 2 ≤ i ≤ n - 1, we get wt_λ(v_iv_i+1) < wt_λ(v_i). Thus, vertex-weights and edge-weights are also pairwise distinct in P_n under the labeling λ, for n ≥ 2. □ Lemma 2.2.Every 1-regular graph is PAT.

Proof. Every 1-regular graph is a nP₂ graph, for n ≥ 1. We fixed the labeling λ of P₂ by λ(v₁) =1, λ(v₂) =2 and λ(v₁v₂) =3. Then we add 3(j - 1) to each labels in j^th P₂ graph, for j ≥ 1. Under this labeling, we get vertex-weighted forms within a sequence {4, 10, 16, …, 4 + 6(n - 1) , 5, 11, …, 5 + 6(n - 1)} and edge-weights forms a sequence {6, 15, …, 6 + 9(n - 1)}. We get our proof from the above sequences. □ Theorem 2.3. The cycle graph C_n is PAT, for n ≥ 3. Proof. Let v₁, v₂, …, v_n be the vertices of a cycle graph C_n, for n ≥ 3 and let λ be the total labeling of C_n with $\begin{matrix} λ (v_{i}) & = & i, for i = 1, 2, \dots, n \\ λ (v_{i} v_{i + 1}) & = & 2 n + 1 - i, for i = 1, 2, \dots, n - 1 \\ λ (v_{1} v_{n}) & = & n + 1 \end{matrix}$ λ enables us to easily conclude that C_n is PAT. □ Lemma 2.4. A disjoint union of two 2-regular PAT graphs is also a PAT graph.

Proof. Let G₁ and G₂ be two 2-regular PAT graphs, and k be the sum of the vertex and edge set cardinality of G₁. Label G₁ as PAT using the first k integers. In order to create a new G₂ PAT labeling, we added k to each label in an old G₂ PAT labeling. The weights of each vertices and edges of G₂ have now been increased by 3k. As a result, the new G₂ labeling contains different vertex and edge weights. Then there’s no conflict in G₁ and none in G₂. In addition, all vertices and edges in G₁ and G₂ have separate weights. Since each vertex and edge weight in G₁ is less than 3k, and each vertex and edge weight in G₂ is greater than 3k. □ Corollary 2.5. If G is a 2 - regular PAT graph, then mG is PAT, for m ≥ 1.

Corollary 2.6. Every simple 2 - regular graph is PAT.

A star graph S_k is a special type of graph in which k vertices have degree 1 and a single central vertex have degree k. It is a complete bipartite graph K_1,k.

Theorem 2.7. The star graph S_k is PAT, for k ≥ 2. Proof. Let v be the central vertex of S_k and v₁, v₂, …, v_k be other vertices of S_k. Let λ be the total labeling of S_k with $\begin{matrix} λ (v) & = & 1 \\ λ (v_{i}) & = & i + 1, for i = 1, 2, \dots, k \\ λ ({vv}_{i}) & = & k + i + 1, for i = 1, 2, \dots, k \end{matrix}$ For 1 ≤ i ≤ k, we get $\begin{matrix} {wt}_{λ} (v) & = & λ (v) + \sum_{i = 1}^{k} λ ({vv}_{i}) \\ = & \frac{1}{2} [3 k^{2} + k + 4] \\ {wt}_{λ} (v_{i}) & = & k + 2 i + 2 \\ < & {wt}_{λ} (v) \\ {wt}_{λ} ({vv}_{i}) & = & k + 2 i + 3 \\ < & {wt}_{λ} (v) \\ {wt}_{λ} (v_{i}) & < & {wt}_{λ} ({vv}_{i}) \end{matrix}$ Thus, S_k is PAT under the labeling λ, for k ≥ 2. □ Bistar graph B(n, n) is the graph obtained by joining the central vertices of two copies of star graph.

Theorem 2.8. Bistar graph B_n,n is PAT, for n ≥ 1.

Proof. Let λ be the total labeling of a bistar graph B_n,n, for n ≥ 1 with $\begin{matrix} λ (u_{i}) & = & i, for i = 1, 2, \dots, n \\ λ (v_{i}) & = & n + i, for i = 1, 2, \dots, n \\ λ (u_{i} u) & = & 2 n + i, for i = 1, 2, \dots, n \\ λ (v_{i} v) & = & 3 n + i, for i = 1, 2, \dots, n \\ λ (u) & = & 4 n + 1 \\ λ (v) & = & 4 n + 2 \\ λ (uv) & = & 4 n + 3 \end{matrix}$ Then λ is a PAT labeling of B_n,n, for n ≥ 1. □ An (n, t)-kite graph consists of a cycle of length n with a t-edge path (the tail) attached to one vertex.

Theorem 2.9. The (n, t)-kite graph is PAT for n ≥ 3 and t ≥ 1.

Proof. Let G be a (n, t)-kite graph, with p and q denoting the number of vertices and edges in G, n being the length of the cycle, and t denoting the number of edges on the path (tail).

Let λ be the total labeling of G and v₁, v₂, …, v_n be vertices of the cycle and w₁, w₂, …, w_t be vertices of the path.

Without loss of generality, we assume that the path is attached to the cycle at v₁. Let $\begin{matrix} λ (v_{i}) & = & i, for i = 1, 2, \dots, n \\ λ (v_{i} v_{i + 1}) & = & 2 n + 1 - i, for i = 1, 2, \dots, n - 1 \\ λ (v_{1} v_{n}) & = & n + 1 \\ λ (w_{j}) & = & 2 n + j, for j = 1, 2, \dots, t \\ λ (v_{1} w_{1}) & = & p + q \\ λ (w_{j} w_{j + 1}) & = & p + q - j, for j = 1, 2, \dots, t - 1 \end{matrix}$ For 2 ≤ i ≤ n - 1 and 2 ≤ j ≤ t, $\begin{matrix} {wt}_{λ} (v_{i}) & = & 4 n + 3 - i \\ {wt}_{λ} (v_{1}) & = & 3 n + p + q + 2 \\ {wt}_{λ} (v_{n}) & = & 3 n + 3 \\ {wt}_{λ} (w_{1}) & = & 2 (p + q + n) \\ {wt}_{λ} (w_{t}) & = & 2 n + p + q + 1 \\ {wt}_{λ} (w_{j}) & = & 2 (p + q + n) - j + 1 \end{matrix}$ As a result of the preceding, we get wt_λ(v_n) < wt_λ(v_i) < wt_λ(w_t) < wt_λ(v₁) < wt_λ(w_j) < wt_λ(w₁) i.e, vertex-weights are pairwise distinct. For 1 ≤ i ≤ n - 1 and 1 ≤ j ≤ t - 1, $\begin{matrix} {wt}_{λ} (v_{i} v_{i + 1}) & = & 2 (n + 1) + i \\ {wt}_{λ} (v_{1} v_{n}) & = & 2 (n + 1) \\ {wt}_{λ} (v_{1} w_{1}) & = & p + q + 2 n + 2 \\ {wt}_{λ} (w_{j} w_{j + 1}) & = & p + q + 4 n + j + 1 \end{matrix}$ From the above equations, we get wt_λ(v₁v_n) < wt_λ(v_iv_i+1) < wt_λ(v₁w₁) < wt_λ(w_jw_j+1) . i.e, edge-weights are pairwise distinct. For 1 ≤ i ≤ n - 1 and 1 ≤ j ≤ t - 1, $\begin{matrix} {wt}_{λ} (v_{1}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}), {wt}_{λ} (v_{1} w_{1}) \\ {wt}_{λ} (v_{1}) & < & {wt}_{λ} (w_{j} w_{j + 1}) \\ {wt}_{λ} (v_{i}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}) \\ {wt}_{λ} (v_{i}) & < & {wt}_{λ} (v_{1} w_{n}), {wt}_{λ} (w_{j} w_{j + 1}) \\ {wt}_{λ} (v_{n}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}) \\ {wt}_{λ} (v_{n}) & < & {wt}_{λ} (v_{1} w_{1}), {wt}_{λ} (w_{j} w_{j + 1}) \\ {wt}_{λ} (w_{1}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}), {wt}_{λ} (v_{1} w_{1}), \\ {wt}_{λ} (w_{j} w_{j + 1}) \\ {wt}_{λ} (w_{t}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}) \\ {wt}_{λ} (w_{t}) & < & {wt}_{λ} (v_{1} w_{1}), {wt}_{λ} (w_{j} w_{j + 1}) \\ {wt}_{λ} (w_{j}) & > & {wt}_{λ} (v_{1} v_{n}), {wt}_{λ} (v_{i} v_{i + 1}), {wt}_{λ} (v_{1} w_{1}), \\ {wt}_{λ} (w_{j} w_{j + 1}) \end{matrix}$ i.e, vertex-weights and edge-weights are also pairwise distinct. Thus the proof. □

3 Dual of PAT graphs

Duality of vertex antimagic total labeling graph is investigated in [2]. Let G be a (p, q)-graph and λ : V(G) ∪ E(G) → {1, 2, …, p + q} represents the total labeling of G. The map on V(G) ∪ E(G) can be defined by . is obviously a one-to-one map from V(G) ∪ E(G) to {1, 2, …, p + q}. Then is a dual of λ.

Theorem 3.1. ([7])Let G be a r-regular graph, r ≥ 1 and let λ and be the total labeling and its dual respectively. Then , ∀v ∈ V(G).

Remark 3.2. Let G be any regular PAT graph. Then the weight of each vertices under its dual are pairwise distinct.

Remark 3.3. Let G be any PAT graph. Then the weight of each edges under its dual are pairwise distinct.

3.1 Dual of some families of PAT graphs

This section shows that the dual of any PAT graph need not be a PAT graph and also lays out the required and sufficient condition for every PAT graph’s dual to be a PAT graph.

Theorem 3.4. Let λ be any total labeling of P_n, for n ≥ 2. If λ admits PAT labeling then its dual admits PAT labeling, for some λ.

Proof. Let λ be the PAT labeling of P_n, for n ≥ 2 as in theorem 2.1. Since p = n and q = n - 1 for path graph. Then its dual will be $\begin{matrix} λ^{'} (υ_{i}) = 2 n - λ (υ_{i}) \\ = 2 n - i, f o r i = 1, 2, ..., n \\ λ^{'} (υ_{i} υ_{i} + 1) = 2 n λ (υ_{i} υ_{i} + 1) \\ = i, f o r i = 1, 2, ..., n - 1. \end{matrix}$ For 2 ≤ i ≤ n - 1, we get $\begin{matrix} ω t_{λ^{'}} (υ_{i}) = 2 n \\ ω t_{λ^{'}} (υ_{n}) = 2 n - 1 \\ ω t_{λ^{'}} (υ_{i}) = 2 n - 1 + i \end{matrix}$ i.e, vertex-weights of lies within a sequence {2n - 1, 2n, 2n + 1, …, 3n - 2}. For 1 ≤ i ≤ n - 1, we get . Here, edge-weights are lies within a sequence {4n - 2, 4n - 3, …, 3n}. i.e, vertex-weights and edge-weights of are pairwise distinct. Hence dual of a path graph P_n under this labeling λ is again a PAT labeling of P_n, for n ≥ 2. □ Corollary 3.5. The above construction of the theorem is true for all λ if and only if it satisfies

wt_λ(v) ≠ wt_λ(w) -(p + q + 1)

wt_λ(v) ≠ wt_λ(e) -(p + q + 1)

where v and w are the vertices of degree 1 and 2 respectively and e ∈ E(P_n).

Proof. Assume that a bijection is PAT. Then for every v, w ∈ V(P_n), e ∈ E(P_n) and n ≥ 2 with v and w are the vertices of degree 1 and 2 respectively

$\begin{array}{l} w t_{λ'} (υ) \neq w t_{λ'} (ω) a n d \\ w t_{λ'} (υ) \neq w t_{λ'} (e) \end{array}$

Conversely, suppose is not PAT. Then for some v, w ∈ V(P_n), e ∈ E(P_n)

$\begin{array}{l} w t_{λ'} (υ) \neq w t_{λ'} (ω) (o r / a n d) \\ w t_{λ'} (υ) \neq w t_{λ'} (e) \end{array}$

which is a contradiction to our assumption. □ Theorem 3.6. Dual of a r-regular PAT graph G is PAT if and only if wt_λ(v) ≠ wt_λ(e) +(r - 2)(p + q + 1), for all v ∈ V(G), e ∈ E(G) and r ≥ 1.

Proof. Assume that a bijection is PAT, where G is a r-regular PAT graph with PAT labeling λ. Then, for every v ∈ V(G) and e ∈ E(G)

$w t_{λ (')} \neq w t_{λ (')} (e)$

wt_λ(v) ≠ wt_λ(e) +(r - 2)(p + q + 1)

Conversely, suppose is not PAT. Since G is r-regular PAT graph, weight of each vertices under is parwise distinct and weight of each edges under is pairwise distinct. Then for some v ∈ G and e ∈ G,

$\begin{matrix} w t_{λ^{'}} (υ) = w t_{λ^{'}} (e) \\ (r + 1) (p + q + 1) - w t_{λ} (v) = 3 (p + q + 1) - w t_{λ} (e) \end{matrix}$

wt_λ(v) = wt_λ(e) +(r - 2)(p + q + 1) which is a contradiction to our premise. □ Corollary 3.7. Dual of a cycle graph C_n is PAT if and only if C_n is PAT, for n ≥ 3.

Proof. Assume that a bijection λ : C_n → {1, 2, …, 2n} is PAT. Suppose its dual is not PAT. Then for some v ∈ V(C_n) and e ∈ E(C_n), $\begin{matrix} w t_{λ (')} (υ) = w t_{λ (')} (e) \\ 3 (2 n + 1) - w t_{λ} (υ) = 3 (2 n + 1) - w t_{λ} (e) \\ w t_{λ} (υ) = w t_{λ} (e) \end{matrix}$ which is a contradiction to that λ is PAT.

Conversely assume that dual of C_n is PAT. Suppose that C_n is not a PAT graph. i.e, there exists for some v ∈ V(C_n) and e ∈ E(C_n) such that $\begin{matrix} w t_{λ} (υ) = w t_{λ} (e) \\ 3 (2 n + 1) - w t_{λ} (υ) = 3 (2 n + 1) - w t_{λ} (e) \\ w t_{λ^{'}} (υ) = w t_{λ^{'}} (e) \end{matrix}$ which is a contradiction that is PAT. □ Corollary 3.8. Dual of a 2-regular PAT grapgh G is PAT if and only if wt_λ(v) ≠ wt_λ(e), for all v ∈ V(G) , e ∈ E(G) and λ is PAT labeling of G.

Corollary 3.9. Dual of a complete PAT graph K_n is PAT if and only if it satisfies wt_λ(v) ≠ wt_λ(e) +(n - 3)(p + q + 1), for all v ∈ V(K_n), e ∈ E(K_n) and n ≥ 1.

Proof. Assume that a bijection is PAT. Then for every v ∈ V(K_n), e ∈ E(K_n) and n ≥ 1

$\begin{matrix} w t_{λ^{'}} (υ) \neq w t_{λ^{'}} (e) \\ n (p + q + 1) - w t_{λ} (υ) \neq 3 (p + q + 1) - w t_{λ} (e) \\ w t_{λ} (υ) \neq w t_{λ} (e) + (n - 3) (p + q + 1) . \end{matrix}$

Conversely suppose is not PAT. Then for some v ∈ V(K_n) and e ∈ E(K_n)

$\begin{matrix} w t_{λ^{'}} (υ) = w t_{λ^{'}} (e) \\ n (p + q + 1) - w t_{λ} (υ) = 3 (p + q + 1) - w t_{λ} (e) \\ w t_{λ} (υ) = w t_{λ} (e) + (n - 3) (p + q + 1) . \end{matrix}$

which is a contradiction to that wt_λ(v) ≠ wt_λ(e) +(n - 3)(p + q + 1). □ Theorem 3.10. Dual of an irregular PAT graph G is PAT if and only if it satisfies

wt_λ(v) ≠ wt_λ(w) -(j - i)(p + q + 1), for 1 ≤ i ≤ j

wt_λ(u) ≠ wt_λ(e) +(k - 2)(p + q + 1), for k ≥ 1

for every v, w, u ∈ V(G), e ∈ E(G) and i, j, k are the degree of the vertices v, w, u respectively.

Proof. Let G be an irregular PAT graph with λ as its PAT labeling. Assume that a bijection is PAT. Then for every v, w, u ∈ V(G), e ∈ E(G) with v, w and u are the vertices of degree i, j and k respectively. Since is a PAT labeling, we have and which implies that (i + 1)(p + q + 1) - wt_λ(v) ≠(j + 1)(p + q + 1) - wt_λ(w) and (k+1)(p+q+1)-wt_λ (u) neq 3(p+q+1)-wt_λ(e), for 1 ≤ i ≤ j and k ≥ 1 $\begin{matrix} i . e, {wt}_{λ} (v) & \neq & {wt}_{λ} (w) - (j - i) (p + q + 1) and \\ {wt}_{λ} (u) & \neq & {wt}_{λ} (e) + (k - 2) (p + q + 1) \end{matrix}$ Conversely, Assume that (i) and (ii) are holds for λ. Suppose is not PAT. Then for some v, w, u ∈ V(G) and e ∈ E(G) $\begin{matrix} w t_{λ^{'}} (υ) = w t_{λ^{'}} (w) (o r / a n d) \\ w t_{λ^{'}} (u) = w t_{λ^{'}} (e) w h i c h i m p l i e s t h a t \\ w t_{λ^{}} (v) = w t_{λ} (w) - (j - i) (p + q + 1) (o r / a n d) \\ w t_{λ^{}} (u) = w t_{λ} (e) + (k - 2) (p + q + 1), \\ f o r 1 \leq i \leq j a n d k \geq 1 \end{matrix}$ which is a contradiction to our assumption. □ Corollary 3.11. Dual of a star PAT graph S_k is again a PAT graph if and only if it satisfies

wt_λ(u) ≠ wt_λ(v) +2(k - 1)(k + 1)

wt_λ(u) ≠ wt_λ(e) +2(k - 2)(k + 1)

wt_λ(v) ≠ wt_λ(e) -2(k + 1)

for every u, v ∈ V(S_k), e ∈ E(S_k) and k ≥ 1, where u and v are the vertices of degree k and 1 respectively.

Corollary 3.12. Dual of a bistar PAT graph B_n,n is again a PAT graph if and only if it satisfies

wt_λ(u) ≠ wt_λ(v) +4n(n + 1)

wt_λ(u) ≠ wt_λ(e) +4(n - 1)(n + 1)

wt_λ(v) ≠ wt_λ(e) -4(n + 1)

for every u, v ∈ V(B_n,n), e ∈ E(B_n,n) and n ≥ 1, where u and v are the vertices of degree n + 1 and 1 respectively.

Corollary 3.13. Dual of a kite PAT graph G is a PAT graph if and only if it satisfies

wt_λ(u) ≠ wt_λ(v) -(p + q + 1)

wt_λ(u) ≠ wt_λ(w) -2(p + q + 1)

wt_λ(v) ≠ wt_λ(w) -(p + q + 1)

wt_λ(u) ≠ wt_λ(e) -(p + q + 1)

wt_λ(w) ≠ wt_λ(e) +(p + q + 1)

for every u, v, w ∈ V(G) and e ∈ E(G), where u, v and w are the vertices of degree 1, 2 and 3 respectively.

4 Conclusion

We addressed the topic of finding perfectly antimagic total labeling of graphs. We showed that the PAT labeling is exist for some family of graphs. Finally, we spotted that necessary and sufficient condition for a duality of PAT graph.

We have compiled a list of open problems that need to be looked at further.

Open Problem 4.1. Determine what conditions are required and adequate for a graph to be PAT.

Open Problem 4.2. If possible, then prove that disjoint union of r-regular PAT graph also be a PAT graph, for r ≥ 3.

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