Consecutive z-index vertex magic labeling graphs

1 Introduction

Lotfi A. Zadeh [10], the inventor of fuzzy sets that have been impressively implemented to clear up the multitudinous real lifestyles choice of difficulties, which is ordinarily erratic, incomplete, precise,uncertain. A fuzzy set is a generalization of a crisp set, in which the elements of the set are divided into membership degrees with an upper bound of 1 and a lower bound of 0. In the fuzzy set, each object’s membership grade isn’t identical as probability, willingly it establishes each object’s belonging grade value that has a single value inside the range of [0, 1]. The basic format of the fuzzy graph and some of its properties expanded through Asriel Rosenfield [6] who looked at fuzzy relations on fuzzy sets in 1975. The notion of connectivity in fuzzy graphs began with Yeh and bang in [7]. L.A.Zadeh affords massive articles on fuzzy sets.

An ordered pair of a set of vertices V and a set of edges E is known as a crisp graph. Additionally, the cardinality of the set of vertices and the set of edges is called the order and size of the graph. A bijection f in a crisp graph G = (V, E) mapping from V ∪ E to N which assigns a completely unique natural number to each vertex and/or edge is known as a Labeling. J.Sedl’aceck [3] launched a new type of graph in 1964 which is a magic graph. He illustrated a graph that is magic when it has an edge-labeling, internal set of real numbers, such that the sum of an edge’s labels and their two end points is equal to a constant. If the sum of the labels related to the vertex is consistent, the impartial of the selection of vertex is called vertex-magic. B.M.Stewart [1] who implemented super magic into the idea of a magic graph. In the fuzzy magic graph, some values for edges or vertices are commonly used. But in fuzzy magic labeling, the vertices and the edges are distinct. Gani et al., who invented the idea of fuzzy labeling and the novel characteristics of fuzzy labeling graphs [2] and fuzzy magic graphs. Mordeson et.al [5] pioneered the notion of “fuzzy labeling" and “fuzzy magic labeling graphs". Additionally, Sunitha [8] utilizes fuzzy bridges and fuzzy cut nodes to gain any identification of fuzzy trees. For simple graphs, Pradeepa pioneered the vertex magic labeling properties in fuzzy graphs and M.Fathalian et.al [4] developed several properties for fuzzy magic labeling. Fuzzy labeling models that receive more attention, recognition, unity to the system, evaluated with the classical and fuzzy models. Most of the authors discuss the various forms of numerous magic graphs.

In this manuscript, after discovering a few properties related to fuzzy consecutive vertex magic labeling and discussing some generalizations of trees in FCVM labeling along with z-index and proving that some fuzzy graphs are FCVM labelings along with z-index. Finally, we providing an application for FCVM labeling in real-time.

2 Results

Throughout this section, we summarize somebasic definitions and results that we require to refer to the readers in what follows [2, 8]. We establish the presence and non-existence of fuzzy consecutive vertex magic labeling graphs. Any fuzzy relation ρ: U → V on fuzzy sets from U into V is defined as the cartesian product of two sets U and V. The intersection of two membership grades of vertex sets σ (U) and σ (V) is defined as the minimum value of those sets i.e.)σ (U) Λσ (V) = min {(σ (U) , σ (V)}.

Definition 2.1. [4] Let the vertex set V be finite and nonempty. Any graph G = (V, E) is claimed to be a fuzzy graph G = (σ, μ) if there exists a bijective pair of functions σ: V → [0, 1] and μ: E → [0, 1] such that membership values of each edge is smaller than or equivalent to the minimum membership values of its distinct two endpoints. i.e.,) μ (uv) ≤ σ (u) Λσ (v) for all u, v ∈ V, and uv ∈ E.

Definition 2.2. [4] A fuzzy labeling G = (σ, μ) in a fuzzy graph is defined as bijective pair of σ: V → (0, 1] and μ : E → (0, 1] in such a way that the membership values are distinguished from the vertices and edges as well as the membership value of each edge is smaller than the minimum membership grades of its distinct two endpoints. i.e.) μ (uv) < σ (u) Λσ (v) for all u, v ∈ V.

Definition 2.3. [4] A fuzzy magic labeling graph is a fuzzy labeling graph such that the magic value for all edges is the same. It is referred to as M. For each uv ∈ E, we have σ (u) + σ (v) + μ (uv) = M lies within (0, 3) (See Fig. 1).

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Fig. 1

Fuzzy magic labeling with M=0.22.

Definition 2.4. A fuzzy vertex magic graph is a fuzzy labeling graph G = (σ, μ) such that the magic value for all vertices is the same. we have, μ (u, v) + σ (v) + μ (v, w) = m (v) ∈ (0.06, 2.94) for all u, v, w ∈ V. It is referred to as m (G) or m.

Definition 2.5. A simple undirected fuzzy labeling graph G = (σ, μ) is said to be fuzzy consecutive vertex magic labeling along with z-index if σ: V → (0, 1] and μ: E → (0, 1] is bijective pair of functions such that the membership values of the vertices and edges from V ∪ E to the consecutive non-integer values z,2z,3z,...(|V| + |E|) z where (|V| + |E|) = N is the total number of vertices and edges and let z → (0, 1] such that choose z = 0.01 for all N with the properties that σ (v) and μ (vu) are distinct for all u, v ∈ V and uv ∈ E; m (v) = σ (v) + ∑_u∈N(v)μ (vu) has a same fuzzy magic value for every u ∈ V where m ∈ (0, 3).

Definition 2.6. In a fuzzy graph, a fuzzy star graph comprises of two sets of vertices U and V with the order of V is 1 and the order of vertex set U does not exceed 1, such that the membership value is 0 for the edges of the consecutive pairs (u _i , u_i+1) and the membership value of edges (v, u _i ) do not exceed 0, where 1 ≤ i ≤ n which is referred to as S_1,n. The membership grades of the set of vertices and edges in the fuzzy star graph are σ (S_1,n) = {v, u₁, u₂, . . . . u _n } and μ (S_1,n) = {vu _i  ; 1 ≤ i ≤ n} (See Fig. 2).

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Fig. 2

Fuzzy star graph S_1,5.

Definition 2.7. A fan in a fuzzy graph is called a fuzzy fan graph F_1,n is defined as a graph K₁ with a label U attaching all the vertices v _j of P _n through the edges e _j such that μ (U, v _i ) >0 and μ (v _i , v_i+1) <1 and |e _j |>1, where 1 ≤ j ≤ n (See Fig. 3).

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Fig. 3

Fuzzy fan graph F_1,4.

Definition 2.8. The comb graph is a graph on 2n vertices and 2n - 1 edges generated by dot product P _n  ⊙ K₁ of the graphs P _n and K₁, connecting all the vertices of the path of length n to exactly one pendant vertex.

Definition 2.9. A sun S _n is the C _n cycle with an end of an edge connected to each vertex of degree one.

Definition 2.10. An E-star graph is a graph on 6 vertices and 5 edges obtained by attaching the 3 pendant edge on vertices with a path of length 3.

Definition 2.11. The paw graph (3-pan graph) is a graph on four vertices and four edges formed attached to an exact one of the vertex of cycle C₃ by joining a vertex of degree one.

Lemma 2.12. A path of odd length P _n and if n ≥ 3 is vertex magic in the fuzzy graph.

Lemma 2.13 A cycle of odd order C _n is vertex magic in the fuzzy graph.

Theorem 2.14. If the graph of magic fuzzy consecutive vertex along with z-index without a vertex isolated, then the fuzzy magic constant m (v) > (|V| + |E|) z for all v ∈ V.

Proof. Assume G has a fuzzy consecutive vertex magic labeling with z-index fuzzy magic constant m = (|V| + |E|) z. There is also a vertex v ∈ V (G) such that σ (v) = (|V| + |E|) z which is contrary to the fact that G has no isolated vertex. If m (v) < (|V| + |E|) z. Then there is a vertex v ∈ V (G) such that σ (v) < (|V| + |E|) z. Since G does not have an isolated vertex,m (v) = σ (v) + ∑_u∈N(v)μ (v, u) which is contrary to choices of successive labels.

Theorem 2.15. No fuzzy magic labeling graph of the vertex consecutive labeling along with z-index has two or more single vertex K₁ or an isolated edge.

Proof. If two or more isolated vertices with a fuzzy magic constant m in the fuzzy consecutive vertex magic labeling graph G along with z-index then there are isolated vertices labeled by v,w,...

So σ (v) = σ (w) = . . . . = m for all v,w ∈V which cannot be valid in the fuzzy labeling. Since the membership values of vertices σ (v) and σ (w) are distinct. Hence G is not consecutive vertex magic labeling in the fuzzy graph along with z-index.

Suppose that the fuzzy consecutive vertex magic labeling graph G along with z-index has an isolated edge vw then σ (v) + μ (v, w) = σ (w) + μ (w, u) = m. Hence σ (v) = σ (w) which is a contradiction, Since G is a fuzzy labeling graph. Hence the fuzzy consecutive vertex magic labeling along with z-index has no isolated edge.

Theorem 2.16. Every fuzzy magic labeling (edge) other than P₃ and C₃ need not be a fuzzy consecutive vertex magic along with z-index.

Proof. Suppose that P₃ has fuzzy magic labeling graph and takes three consecutive vertices labeled by u,v,w into consideration. Then m (u) = σ (u) + μ (uv) and m (w) = σ (w) + μ (vw) and σ (u) + σ (v) + μ (uv) = M (uv); M (vw) = σ (v) + σ (w) + μ (vw). All the membership values of edges have the same fuzzy magic value since for fuzzy magic. Therefore,

M ( uv ) = M ( vw ) σ ( u ) + σ ( v ) + μ ( uv ) = σ ( v ) + σ ( w ) + μ ( vw ) ; σ ( u ) + μ ( uv ) = σ ( w ) + μ ( vw ) ; m ( u ) = m ( w ) .

In P₃, m (0.05) =0.06; m (0.03) =0.06; m (0.04) =0.06 and M (0.01) =0.09; M (0.02) =0.09. Here all the vertices have the fuzzy magic constant value 0.06 and edges with fuzzy magic value 0.09 [see Fig. 4]. Hence, P₃ is both fuzzy magic and fuzzy consecutive vertex magic labeling along with z-index.

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Fig. 4

Fuzzy magic and vertex magic labeling in P₃ and C₃.

Suppose that C₃ has fuzzy graph of magic labeling and three consecutive vertices labeled by u,v,w. Then m (u) = σ (u) + μ (uv) + μ (vw) and m (v) = σ (v) + μ (vw) + μ (uv) and m (w) = σ (w) + μ (uw) + μ (vw) and also M (uv) = σ (u) + σ (v) + μ (uv); M (uw) = σ (u) + σ (w) + μ (uw); M (vw) = σ (v) + σ (w) + μ (vw). Since for fuzzy magic, every membership values of edges have the same fuzzy magic value.

Therefore,

M (uv) = M (vw) = M (uw) σ (u) + σ (v) + μ (uv) = σ (v) + σ (w) + μ (vw) = σ (u) + σ (w) + μ (uw) Equating each pairs we get,

σ ( u ) + μ ( uv ) = σ ( w ) + μ ( vw ) ; σ ( v ) + μ ( uv ) = σ ( w ) + μ ( uw ) ; σ ( v ) + μ ( vw ) = σ ( u ) + μ ( uw ) .

We conclude that m (u) = m (v) = m (w). In C₃, m (0.06) =0.09; m (0.05) =0.09; m (0.04) =0.09 and M (0.03) =0.12; M (0.02) =0.12; m (0.01) =0.12. Here all the vertices have the fuzzy magic constant value 0.09 and edges with fuzzy magic value 0.12 [see Fig. 4]. Hence, C₃ is both fuzzy magic and fuzzy vertex consecutive magic labeling along with z-index.

Hence path of length 3 and a 3-cycle graph which is both fuzzy magic and fuzzy consecutive vertex magic labeling along with z-index.

Theorem 2.17. In the fuzzy labeling graph, a fuzzy fan graph F_1,n is fuzzy consecutive vertex magic labeling along with z-index if and only if n=2 (See Fig. 5).

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Fig. 5

Fuzzy fan graph F_1,2 with m=0.09.

Proof: Suppose there exists FCVM labeling for fuzzy fan graph n ≥ 2 with magic constant k. Let u be the vertex of degree n.

Assume that the edges incident with u receive the minimum labels, with the possibility of remaining distinct Membership values do not generate the same magic constant value, and the vertex label at u is higher that is greater than or equal to the minimal vertex label, which is a contradiction to the magic constant. As a result, fuzzy fan graph F_1,n is not fuzzy consecutive vertex magic labeling along with z-index if n ≥ 2.

Theorem 2.18. Any graph on four connected vertices is not a fuzzy consecutive vertex magic labeling along with z-index. But the removal of one edge from K₄ is fuzzy consecutive vertex magic labeling along with z-index.

Proof. Assume a fuzzy consecutive vertex magic labeling graph G along with z-index fuzzy magic constant m exists. Here, |V (G) |=4 and four vertices of non-isomorphic connected graphs whose size (See Fig. 6),

| E ( G ) | = | V | - 1 , | V | , | V | + 1 , | V | + 2 . = 3 , 4 , 5 , 6 .

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Fig. 6

Non-isomorphic connected graphs on 4 vertices.

We can easily conclude that path of length 4 and cycle of order 4 is not fuzzy consecutive vertex magic labeling along with z-index. If |E|=6, possible fuzzy consecutive vertex magic labeling along with z-index does not generate the same fuzzy magic constant for all vertices which are incompatible with the fact that distinct membership values for the vertices and edges and also m (v) > (|V| + |E|) z for all v ∈ V. The removal of one edge from K₄ is FCVM labeling along with z-index whose magic constant is 0.16 and K₅ is FCVM labeling with z-index whose m=0.35 shown in Fig. 7.

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Fig. 7

Fuzzy consecutive vertex magic labeling of K₄ - e with m=0.16 and K₅ with m=0.35.

Note 2.19. The complete graph K _n admits FCVM labeling along with z- index iff n is odd.

The purpose of this section is to establish some properties of fuzzy consecutive vertex magic total labeling of graphs along with z-index in trees and generalizations of trees.

Theorem 3.1. In the fuzzy labeling graph G = (σ, μ), the order of G is even and the size of G is equal to |V (G) |-1 or |V (G) |. Then G is not a fuzzy consecutive vertex magic labeling along with z-index.

Proof. Let us consider the even the number of vertices. If the number of edges is |V|-1 then the graph must be the path and if the number of edges and the number of vertices are equal then the graph must be cycle graph.

By lemma 2 and lemma 2 shows that P _n is not fuzzy consecutive vertex magic labeling along with z-index if n is even and C _n is not fuzzy consecutive vertex magic labeling along with z-index if n is even.

Corollary 3.2. Any tree of odd order is Fuzzy consecutive vertex magic along with z-index.

Proof. Consider a tree of even order which is fuzzy consecutive vertex magic labeling along with z-index if n ≥ 2.

If n = 2, the tree consists of exactly one isolated edge which is not FCVM labeling that contradicts the theorem 2. If n = 4, P₄ is not FCVM labeling along with z-index by lemma 2.

Let G be a tree of even order n ≥ 6 and consider one pendent vertex v in G. Then u is a only one neighbourhood of v in some deg (u) ≥2.

Let e₁, e₂, … e_deg(u) be edges incident with u where e₁ = uv. Since G admits fuzzy consecutive vertex-magic labeling along with z-index then k = σ (v) + μ (e₁) = σ (u) + μ (e₁) + μ (e₂) + … μ (e_deg(u)).

Rearranging, we get σ (v) - σ (u) = μ (e₂) + … μ (e_deg(u))

The terms on the right side must all be greater than n, while the terms on the left side must all be less than n which is clearly a contradiction (Fig. 8). Thus tree of even order does not have fuzzy consecutive vertex-magic along with z-index. Hence every tree of odd order is Fuzzy consecutive vertex magic along with z-index.

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Fig. 8

Fuzzy consecutive vertex magic labeling of even order.

For a tree T, |V| = |E|+1. As a consequence, the fuzzy magic constant m = ( 5 v - 3 2 ) z . Since the path of odd length is fuzzy consecutive vertex magic along with z-index. Tree of order 5 with m=0.11 is a fuzzy consecutive vertex magic labeling along with z-index shown in Fig. 9.

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Fig. 9

Fuzzy consecutive vertex magic labeling of tree of order 5 with m=0.11.

Corollary 3.3. The paw graph (3-pan graph) and the E-star graph are not FCVM labeling along with z-index.

Proof. Suppose the paw graph (3-pan graph) is FCVM labeling along with z-index. These must be the label from 1 to 8 but the same fuzzy magic constant is not created by fuzzy consecutive vertex magic labeling along with z- index for all vertices that conflict with a truth that the vertices and edges have distinct membership values for the vertices and the edges, as well as m (v) > (|V| + |E|) z for all v ∈ V. Hence the paw graph (3-pan graph) is not FCVM labeling along with z-index.

The E-star graph is not FCVM labeling along with z-index. Since by corollary [3.2] and the vertices of E-star is even.

Corollary 3.4. Sun graph is not fuzzy consecutive vertex magic labeling along with z-index for all n.

Proof. Suppose we consider the sun graph is FCVM along with z- index for all n.

Let us consider the vertex v has degree 2. Then the neighbourhood of v has a vertex u of degree ≥4 and v₁, v₂, … v_2n-2 are the remaining vertices of sun graph.

The vertex u incident with the edges are x, e₁, e₂, … e_deg(u)-1, where x = vu. Then w (v) = σ (v) + μ (x) + μ (2n - 1) w (u) = σ (u) + μ (x) + μ (e₁) + … μ (e_deg(u)-1)

Then the magic constant m = σ (v) + μ (x) + μ (2n - 1) = σ (u) + μ (x) + μ (e₁) + … μ (e_deg(u)-1) ⇒σ (v) + μ (2n - 1) = σ (u) + μ (e₁) + … μ (e_deg(u)-1) which is not possible because the right hand side term must be greater than n and the left hand side term always less than n. Moreover, the order and size of sun graph are even. By Theorem [3.1], Sun graph is not fuzzy consecutive vertex magic labeling along with z-index for all n.

Corollary 3.5. If n is odd and |E| > |V|, Generalized Peterson graph P(n,m) is not fuzzy consecutive vertex magic along with z-index.

Proof. Suppose there exists a fuzzy consecutive vertex magic labeling of P (n, m) along with z-index with magic constant k if n is even.

By Theorem [3.1] the order of G is even and the size of G is equal to |V (G) |-1 or |V (G) |. Then G is not a fuzzy consecutive vertex magic labeling along with z-index. But here |E| > |V|.

Let G be a P (n, m) graph with set of vertices {v₁, v₂, … v _n , u₁, u₂, … u _n } and edges {v₁v₂, … v _n v₁, v₁u₁, … v _n u _n , u₁u₃, … u _i u_i+m}.

If n is even and n ≥ 4 and 1 ≤ m ≤ n 2 - 1 , P (n, m) has a fuzzy consecutive vertex-magic labeling along with z-index defined by the bijective mapping and we construct the following labelings:

w ( v i ) = { ( 4 + 7 n ) z 2 , ( 6 + 7 n ) z 2 , … ( 2 + 9 n ) z 2 } and w ( u i ) = { ( 4 + 9 n ) z 2 , ( 6 + 9 n ) z 2 , . . . . ( 2 + 11 n ) z 2 } are the set of weights of all vertices in P (n, m).

The edge labelings is as follows:

μ ( v i v i + 1 ) = { ( n - i 2 + 1 ) zifiiseven z 2 + i + 5 n 2 ifiisodd μ ( u i u i + m ) = { 4 nz + 6 z - iz 2 ifiiseven & i ≤ 4 ( 5 n + 6 ) z 2 - zi 2 ifiiseven & i ≥ 6 zi 2 + ( 1 + 3 n ) z 2 ifiisodd μ ( v i u i ) = { 2 nz + i - 1 2 ifiiseven & i ≤ 4 ( n + i ) z 2 - 2 zifiiseven & i ≥ 6 ( 1 + 3 n ) z 2 - zi 2 ifiisodd

As a consequence, the weights of vertices of the P (n, m) take up the set of consecutive integers { ( 4 + 7 n ) z 2 , ( 6 + 7 n ) z 2 , … ( 2 + 9 n ) z 2 ( 4 + 9 n ) z 2 , … , ( 2 + 11 n ) z 2 } .

Here n = 6, The generalized peterson graph on 12 vertices admits FCVM labeling along with z-index and its fuzzy magic constant is m=0.53 [see Fig. 10].

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Fig. 10

Generalized Peterson graph P_6,2 with m=0.53.

Hence for any generalized Peterson graph P (n, m) admits fuzzy consecutive vertex magic along with z-index if n is even and m lies between 1 and n 2 - 1 and it has fuzzy magic constant m = ( 17 n 2 + 2 ) z . Hence we can easily conclude that P (n, m) is not fuzzy consecutive vertex-magic labeling along with z-index if n is odd and |E| > |V|.

Theorem 3.6. Not every complete bipartite graph is the fuzzy consecutive vertex magic labeling along with z-index except K_1,2.

Proof. Consider any two consecutive vertices u and v in the fuzzy consecutive vertex magic labeling graph along with z-index. So the fuzzy magic constant for both vertices is the same which is only possible because both vertices have the same membership values which are contradictory to the fact that G is a fuzzy labeling graph. Then K_1,1 is not fuzzy consecutive vertex magic along with z-index and suppose that the fuzzy labeling graph G has three consecutive vertices labeled with u,v,w.

Then m (u) = m (v) = m (w) which is feasible only when σ (u) = σ (v) + μ (v, w) ; σ (w) = σ (v) + μ (u, v). Since σ (u) + μ (u, v) = σ (w) + μ (v, w) = σ (v) + μ (u, v) + μ (v, w) and also the membership values of vertices and edges σ (u), σ (v), σ (w) and μ (u, v), μ (v, w), μ (v, w) are all distinct and σ (v) + μ (u, v) + μ (v, w) =0.06 is identical for each vertex. Hence K_1,2 is fuzzy consecutive vertex magic along with z-index (see Fig. 11) and also we say that no fuzzy star labeling graph except S_1,2 is fuzzy consecutive vertex magic along with z-index fuzzy magic constant 0.06.

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Fig. 11

Fuzzy consecutive vertex magic labeling of K_1,2 with m=0.06.

Corollary 3.7. No complete bipartite graph hamiltonian is a fuzzy consecutive vertex magic along with z-index.

Proof. Since for any complete bipartite K_m,n is Hamiltonian iff m = n. If m = 1 in K_m,n, K_1,1 ≃ P₂, which is not fuzzy consecutive vertex magic along with z-index. All the other complete bipartite graph hamiltonian is not fuzzy consecutive vertex magic along with z-index which contradicts the fact that the membership value of the distinct vertices and edges of the consecutive values and m > (|V| + |E|) z.

Theorem 3.8. Let G be a graph formed by joining the endpoint of degree 2 of a comb graph to a pendant vertex admits fuzzy consecutive vertex magic labeling along with z-index.

Proof. We defined the set of vertices and edges of a comb graph are {u₁, u₂ . . . . u _t } ∪ {u₁₁, u₂₁, u₃₁, . . . u_t1} and {u₁u₁₁, u₂u₂₁, . . . u _t u_t1} ∪ {u _i u_i+1 : 1 ≤ i ≤ t - 1} respectively.

Let G be a graph formed by joining the endpoint of degree 2 of a comb graph to a pendant vertex with set of vertices {u₁, u₂ . . . . u _t } ∪ {u₁₁, u₁₂, u₂₁, u₃₁, . . . u_t1} and edges {u₁u₁₁, u₁u₁₂, u₂u₂₁, . . . u _t u_t1} ∪ {u _i u_i+1 : 1 ≤ i ≤ t - 1}.

The vertex labelings as follows:

σ ( u i 1 ) = ( 4 t + 1 - i ) zif 1 ≤ i ≤ t σ ( u i ) = ( 2 t + i ) zif 1 ≤ i ≤ t σ ( u 12 ) = ( 4 t + 1 ) z The edge labelings as follows:

μ ( u i u i + 1 ) = ( t - i ) zif 1 ≤ i ≤ t - 1 μ ( u i u i 1 ) = ( t + i ) zif 1 ≤ i ≤ t μ ( u 1 u 12 ) = tz

The magic values for initial, terminal and intermediate vertices as follows:

m ( u 12 ) = σ ( u 12 ) + μ ( u 1 u 12 ) = ( 4 t + 1 ) z + tz = ( 5 t + 1 ) z m ( u 1 ) = σ ( u 1 ) + μ ( u 1 u 12 ) + μ ( u 1 u 2 ) + μ ( u 1 u 11 ) = ( 2 t + 1 ) z + tz + ( t - 1 ) z + ( t + 1 ) z = ( 5 t + 1 ) z m ( u t ) = σ ( u t ) + μ ( u t - 1 u t ) + μ ( u t u t 1 ) = 3 tz + z + 2 tz = ( 5 t + 1 ) z m ( u i 1 ) = σ ( u i 1 ) + μ ( u i u i 1 ) , 1 ≤ i ≤ t = ( 4 t + 1 - i ) z + ( t + i ) z = ( 5 t + 1 ) z m ( u i + 1 ) = σ ( u i + 1 ) + μ ( u i u i + 1 ) + μ ( u i + 1 u i + 2 ) + μ ( u i + 1 u ( i + 1 ) 1 ) , 1 ≤ i ≤ t - 2 = ( 2 t + i + 1 ) z + ( t - i ) z + ( t - i - 1 ) z + ( t + i + 1 ) z = ( 5 t + 1 ) z

Here t=7, The endpoint of degree 2 of a comb graph on 14 vertices attaching to exactly one pendant vertex u₁₂ admits FCVM labeling along with z-index and its fuzzy magic constant is m=0.36 (See Fig. 12).

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Fig. 12

Fuzzy consecutive vertex magic labeling of graph with magic value m=0.36.

Hence for any graph formed by joining the endpoint of degree 2 of a comb graph to exactly one pendant vertex admits fuzzy consecutive vertex magic along with z-index and it has fuzzy magic constant m = (5t + 1) z.

In this section, we explore some results on families of graphs like Torus graph, Prism graph, Hypercubes, Parachuate graph and and Generalized (Stacked) prism graph, Generelized butterfly graph are fuzzy consecutive vertex magic along with z-index.

The weighted fuzzy vertex magic sum,

m (v _i ) = σ (v _i ) + ∑_{v j ∈N(v i )}μ (v _i , v _j ) , 1 ≤ i ≤ n and v _i  ∈ V.

4.1 Torus graph

A Torus graph T_n,n, (n ≥ 3) is a graph on n ² vertices and 2n ² edges obtained from the graph cartesian product T_n,n = C _n  × C _n formed by connecting all the vertices of the cycle graphs C _n .

The graph T_3,3 has fuzzy consecutive vertex magic along with z-index with magic value 0.61 which is shown in Fig. 13.

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Fig. 13

Torus T_3,3 with magic value m=0.61.

If n is odd, The fuzzy magic constant of T_n,n for fuzzy consecutive vertex magic labeling along with z-index is ( 17 n 2 + 5 2 ) z .

4.2 Prism graph

A prism graph D _n , (n ≥ 3) is given by the cartesian product D _n  = C _n  × P₂ formed by connecting n concentric cycle graph C _n to P₂. It is an undirected 3-regular graph on 2n vertices and 3n edges.

The graph D₆ has fuzzy consecutive vertex magic along with z-index with magic value 0.53 which is shown in Fig. 14.

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Fig. 14

Fuzzy consecutive vertex magic of D₆ with magic value m=0.53.

If n is even and n ≥ 4, The fuzzy magic constant of D _n for fuzzy consecutive vertex magic labeling along with z-index is ( 23 n + 4 2 ) z .

4.3 Hybercube graph

A hypercube graph Q _n is the cartesian product of nK₂ copies of an n-dimensional hypercube. It is an undirected n-connected regular graph and is obtained from the vertices 2 ⁿ and edges n2^n-1.

The graph Q₁ consists of 2 vertices and 1 edge of a 1-dimensional hypercube and Q₁ is isomorphic to P₂, which is not fuzzy consecutive vertex magic labeling along with z-index and the graph Q₂ comprises of 4 vertices and 4 edges of a 2-dimensional hypercube and Q₂ ≅ P₄, which is also not consecutive vertex magic labeling along with z-index in fuzzy graphs. If n=3, the cubical graph Q₃ is a 3-regular graph with 8 vertices and 12 edges, which is fuzzy consecutive magic labeling along with z-index (See Fig. 15).

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Fig. 15

Fuzzy consecutive vertex magic of Q₃ with magic value m=0.36.

If n is odd and n ≥ 3, The fuzzy magic constant of hypercube Q _n for fuzzy consecutive vertex magic labeling along with z-index is m = ( n 2 ( 2 2 n - 2 ) + n ( 2 n - 1 ) 2 n + 2 n ( n + 1 ) - 2 n + 1 2 + 1 ) z .

4.4 Parachuate graph

Parachuate graph Pr _n , (n ≥ 3) is a connected irregular graph on 2n and 3n - 1, obtained by joining the two endpoints of the path to the fan graph F_1,n with the path of length n - 1. It has a minimum degree of 2 and a maximum degree n.

The fuzzy magic constant of Pr _n for fuzzy consecutive vertex magic labeling along with z-index m = ( 17 n - 4 2 ) z arise only if n is even and n ≥ 4 (See Fig. 16).

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Fig. 16

Pr₄ with fuzzy consecutive vertex magic value m=0.32.

5 Illustrated example for fuzzy consecutive vertex magic labeling

In this section, we establish the application for fuzzy consecutive vertex magic labeling along with z-index in real-time problems.

Problem 5.1 An organization data set represents a game in 2021. That organization has a desire to offers an equal number of games to its 12 teams. That organization allowed three games to play by teams. When each team playing three games in various teams, Calculate the percentage of games allocated to each team. Determine the proportion of games played by teams only by treating the graph admits a fuzzy consecutive vertex labeling along with z-index (See Fig. 18).

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Fig. 18

A regular graph with 12 vertices and 18 edges.

Solution:

It should be remembered that in an undirected network, Consider the following arbitrary network of 12 nodes A network node represents a team, and the link represents a game between two teams. The network’s group composition is well known: Games between the teams are getting more popular. The membership values of the teams are σ (v₁), σ (v₂), σ (v₃), σ (v₄), σ (v₅), σ (v₆), σ (v₇), σ (v₈), σ (v₉), σ (v₁₀),σ (v₁₁), σ (v₁₂) respectively and we take an edge between teams σ (v₁) and σ (v₂) when team 1 and team 2 are shares games. First, we can define the node and edge weights for the network.

Each team is a node in the network’s weighted graph. The team’s core degree in the network is represented by the node weight. The attraction between teams is the edge weight. Yet teams are assumed to have the same weight, so the weights are still the same value link will occur for any nodes. We propose a fuzzy consecutive labeling, which can increase the weight of the network and enhance network capacity.

As a result, to improve network efficiency, the capacities of the nodes must be increased. Stability of weighted sum can be achieved by changing the labels in an appropriate manner.

A=

[ 0.23 0 0.01 0 0 0 0.04 0.09 0 0 0 0 0 0.22 0 0.06 0.11 0 0 0 0.08 0 0 0 0.01 0 0.30 0 0.16 0 0 0 0.03 0 0 0 0 0.06 0.16 0.25 0.17 0 0 0 0 0 0 0 0 0.11 0 0 0.24 0 0 0.12 0 0 0 0.07 0 0 0 0.17 0 0.28 0 0 0 0.02 0 0.10 0.04 0 0 0 0 0 0.26 0 0 0 0.14 0.18 0.09 0.08 0 0 0.12 0 0 0.29 0 0 0 0 0 0 0.03 0 0 0 0 0 0.19 0.05 0.15 0 0 0 0 0 0 0.02 0 0 0.05 0.20 0.13 0 0 0 0 0 0 0 0.14 0 0.15 0.13 0.27 0 0 0 0 0 0.07 0.10 0.18 0 0 0 0 0.21 ] m (v₁) =0.23 + 0.01 + 0.09 + 0.04 = 0.37

m (v₂) =0.22 + 0.08 + 0.11 + 0.06 = 0.47

m (v₃) =0.30 + .01 + 0.03 + 0.16 = 0.50

m (v₄) =0.25 + 0.16 + 0.06 + 0.17 = 0.64

m (v₅) =0.24 + 0.07 + 0.12 + 0.11 = 0.54

m (v₆) =0.28 + 0.02 + 0.10 + 0.17 = 0.57

m (v₇) =0.26 + 0.18 + 0.14 + 0.04 = 0.62

m (v₈) =0.29 + 0.09 + 0.12 + 0.08 = 0.58

m (v₉) =0.19 + 0.05 + 0.03 + 0.15 = 0.42

m (v₁₀) =0.20 + 0.13 + 0.05 + 0.02 = 0.40

m (v₁₁) =0.27 + 0.14 + 0.13 + 0.15 = 0.69

m (v₁₂) =0.21 + 0.18 + 0.10 + 0.07 = 0.56

The link from the ith team to the jth team is a link for each entry in A. Here teams have a distinct weight, so the weighted sums are not the same value flow for any nodes. Next, we consider a structure. To achieve this, we need not change the topology of the network. Order to achieve our goal of having the same weight of all nodes in a network can be achieved by appropriately changing the labels and if the network is not partially damaged, the migration to other regions would be unaffected. A new structure termed fuzzy consecutive vertex- magic labeling is proposed

B=

[ 0.27 0 0.16 0 0 0 0.01 0.09 0 0 0 0 0 0.24 0 0.08 0.11 0 0 0.10 0 0 0 0 0.16 0 0.29 0.03 0 0 0 0 0.05 0 0 0 0 0.08 0.03 0.25 0 0.17 0 0 0 0.06 0 0 0 0.11 0 0 0.23 0 0 0.12 0 0 0 0.07 0 0 0 0.17 0 0.28 0 0 0 0.06 0 0.02 0.01 0 0 0 0 0 0.30 0 0 0 0.04 0.18 0.09 0.10 0 0 0.12 0 0 0.22 0 0 0 0 0 0 0.05 0 0 0 0 0 0.19 0.14 0.15 0 0 0 0 0 0 0.06 0 0 0.14 0.20 0.13 0 0 0 0 0 0 0 0.04 0 0.15 0.13 0.21 0 0 0 0 0 0.07 0.02 0.18 0 0 0 0 0.26 ]

The weighted sum,

m (v _i ) = σ (v _i ) + ∑_{v j ∈N(v i )}μ (v _i , v _j ) , 1 ≤ i ≤ n and v _i  ∈ V.

The link from the ith team to the jth team is a link for each entry in B. The undirected graph’s fuzzy labeling matrix A is rebuilt with a change in the labeling of both vertices and edges to apply the methodology elaborated in this paper.

The vertex labels of resultant graph,

σ ( v 1 ) = 0.27 ; σ ( v 7 ) = 0.30 σ ( v 2 ) = 0.24 ; σ ( v 8 ) = 0.22 σ ( v 3 ) = 0.29 ; σ ( v 9 ) = 0.19 σ ( v 4 ) = 0.25 ; σ ( v 10 ) = 0.20 σ ( v 5 ) = 0.23 ; σ ( v 11 ) = 0.21 σ ( v 6 ) = 0.28 ; σ ( v 12 ) = 0.26 The edge labelings of resultant graph,

μ ( e 1 ) = 0.09 ; μ ( e 2 ) = 0.05 μ ( e 3 ) = 0.08 ; μ ( e 4 ) = 0.06 μ ( e 5 ) = 0.07 ; μ ( e 6 ) = 0.04 μ ( e 7 ) = 0.16 ; μ ( e 8 ) = 0.03 μ ( e 9 ) = 0.17 ; μ ( e 10 ) = 0.02 μ ( e 11 ) = 0.18 ; μ ( e 12 ) = 0.01 μ ( e 13 ) = 0.10 ; μ ( e 14 ) = 0.12 μ ( e 15 ) = 0.14 ; μ ( e 16 ) = 0.15 μ ( e 17 ) = 0.11 ; μ ( e 18 ) = 0.13 m (v₁) =0.27 + 0.16 + 0.09 + 0.01 = 0.53

Similarly, m (v _i ) =0.53 for all v _i  ∈ B, i=1,2, ... 12

The fuzzy magic value m for one team represents the weighted sum of games. The exact number of games played by one team indicated by the vertex labels σ (v _i ), i = 1, 2, …, .12 Here m = 0.53 (See Fig. 19). Hence every team affords the number percentage of 0.53% of games. The proportion of games for a specific team is a diagonal entry of FCVM matrix B.

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Fig. 19

Fuzzy consecutive vertex magic labeling of graph with m=0.53.

The ultimate main intend of fuzzy models is to diminish the actual error value in models that can not be used exhaustively in several fields with long-established mathematical models, especially in the fields of scientific modeling, telecommunications and so on. In this paper, we introduced a new type of fuzzy labeling, named “Fuzzy consecutive vertex magic labeling graphs” with z-index fuzzy magic constant and revealed the presence of a notion of tree in consecutive vertex labeling fuzzy graphs with z-index and certain families of fuzzy graphs are proved to be fuzzy graphs of consecutive vertex labeling graphs along with z-index. Also we provide an application for the consecutive vertex labeling along with z-index in a fuzzy magic graph. Finally, we would like to conclude the paper with a very significant hypothesis. More findings in this field, as well as their implementations, will be addressed in future papers.

Conjecture 6.1. For any positive integers t, K _m  ⋃ P _m is fuzzy consecutive vertex magic when m = 8t + 2.