SEMT valuation and strength of subdivided star of K
               1,4

Salma Kanwal; Mariam Imtiaz; Nazeran Idrees; Zurdat Iftikhar; Tahira Sumbal Shaikh; Misbah Arshad; Rida Irfan

doi:10.1515/math-2020-0067

Open Access Published by De Gruyter Open Access October 20, 2020

SEMT valuation and strength of subdivided star of K _1,4

Salma Kanwal , Mariam Imtiaz , Nazeran Idrees , Zurdat Iftikhar , Tahira Sumbal Shaikh , Misbah Arshad and Rida Irfan

From the journal Open Mathematics

https://doi.org/10.1515/math-2020-0067

Abstract

This study focuses on finding super edge-magic total (SEMT) labeling and deficiency of imbalanced fork and disjoint union of imbalanced fork with star, bistar and path; in addition, the SEMT strength for Imbalanced Fork is investigated.

Keywords: SEMT labeling; SEMT deficiency; SEMT strength; imbalanced fork

MSC 2010: 05C78

1 Preliminaries

Labeling is a technique that allots labels to the components of a graph. Total labeling gives us both components (vertices and edges) labeled. A ( ν , ε ) -graph G determines an edge-magic total (EMT) labeling when Γ : V ( G ) ∪ E ( G ) → { 1 , ν + ε ¯ } is bijective so as the weights at every edge are the same constant (say) c, such a number c is interpreted as a magic constant. If all vertices gain the smallest of the labels, then an EMT labeling is called a super edge-magic total (SEMT) labeling. Kotzig and Rosa [1] and Enomoto et al. [2] found the concepts of EMT and SEMT graphs, respectively, and presented the conjectures: every tree is EMT [1] and every tree is SEMT [2].

If a graph G allows at least one SEMT labeling, then the smallest of the magic constants for all possible distinct SEMT labelings of G describes SEMT strength, sm ( G ) , of G. Avadayappan et al. first introduced the notion of SEMT strength [3] and found the exact values of SEMT strength for some graphs.

In [1], the notion of EMT deficiency was proposed, and Figueroa-Centeno et al. [4] continued it to SEMT graphs. For any graph G, the SEMT deficiency, signified as μ s ( G ) , is the least number n of isolated vertices that we have to take in union with G so that the resulting graph G ∪ n K 1 is SEMT, and the case + ∞ will arise if no isolated vertex fulfills this criterion. More specifically,

μ s ( G ) = min M ( G ) if M ( G ) ≠ ∅ , + ∞ if M ( G ) = ∅ ,

where M ( G ) = { n ≥ 0 : G ∪ n K 1 is an SEMT graph } .

In [4,5], Figueroa-Centeno et al. proposed a conjecture about the confined deficiencies of the forests. In [6], an assumption was made as a special case of a previous one that says, the deficiency of each two-tree forest is not more than 1. The results in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] are found to be useful in the aspect of examined labeling here. For more review, see the recent survey of graph labelings by Gallian [23].

In this paper, we formulated the results on SEMT labeling and deficiency of forests consisting of imbalanced fork and disjoint union of imbalanced fork with star, bistar and path, respectively. The values of parameters of the star, bistar and path are totally dependent on the parameters involved in the imbalanced fork.

2 Main results

Definition 2.1

An imbalanced fork, represented as Fr ( l , l + 1 , l + 2 ) , l ∈ N , see Figure 1, is also a tree consisting of three paths of lengths l, l + 1 and l + 2 , that is,

P l : x 1 , ȷ ; 1 ≤ ȷ ≤ l ,

P l + 1 : x 2 , ȷ ; 1 ≤ ȷ ≤ l + 1 ,

P l + 2 : x 3 , ȷ ; 1 ≤ ȷ ≤ l + 2 .

A single new vertex x 2 , 0 is added to the path P l + 1 through an edge, and these three paths are joined together by two edges that are x ı , 1 x ı + 1 , 1 , 1 ≤ ı ≤ 2 . Precisely, the set of vertices and the set of edges of imbalanced fork are as follows:

V ( Fr ( l , l + 1 , l + 2 ) ) = { x 2 , 0 } ∪ { x ı , ȷ : ı = 1 , 1 ≤ ȷ ≤ l } ∪ { x ı , ȷ : ı = 2 , 1 ≤ ȷ ≤ l + 1 } ∪ { x ı , ȷ : ı = 3 , 1 ≤ ȷ ≤ l + 2 } ,

E ( Fr ( l , l + 1 , l + 2 ) ) = { x ı , ȷ x ı , ȷ + 1 : ı = 1 , 1 ≤ ȷ ≤ l − 1 } ∪ { x ı , ȷ x ı , ȷ + 1 : ı = 2 , 1 ≤ ȷ ≤ l } ∪ { x ı , ȷ x ı , ȷ + 1 : ı = 3 , 1 ≤ ȷ ≤ l + 1 } ∪ { x ı , 1 x ı + 1 , 1 : 1 ≤ ı ≤ 2 } ∪ { x 2 , 0 x 2 , 1 } ,

respectively.

Note 1. Another way of writing imbalanced fork Fr ( l , l + 1 , l + 2 ) is T ( 1 , l , l , l + 2 ) , l ∈ N because of its extraction from a star by its subdivision. In our another paper [24], we have established SEMT labeling and strength, for all positive integers ≥ 2 , of fork which is also a subdivision of star K 1 , 4 . Imbalanced Fork is basically a special case of the Fork.

The following lemma is an elementary tool for proving graphs to be SEMT. It will be used as a base in each result presented in this work.

$Figure 1 Imbalanced Fork Fr ( 4 , 5 , 6 ) \text{Fr}(4,5,6) .$

Figure 1

Imbalanced Fork Fr ( 4 , 5 , 6 ) .

Lemma 2.2

[25] A ( ν , ε ) -graph G is SEMT if and only if ∃ a bijective map Γ : V ( G ) → { 1 , ν ¯ } s.t. the set of edge-sums

S = { Γ ( l ) + Γ ( m ) : l m ∈ E ( G ) }

constructs ε consecutive Z + . In that case, G can extend to an SEMT labeling of G with magic constant c = ν + ε + min ( S ) and

S = { c − ( ν + ε ) , c − ( ν + ε ) + 1 , … , c − ( ν + 1 ) } .

The following result of SEMT graphs also holds.

Note 2. [3] Let c ( Γ ) be a magic constant of an SEMT labeling Γ of G ( V , E ) , then we end up on this statement:

(1) ε c ( Γ ) = ∑ v ∈ V deg G ( v ) Γ ( v ) + ∑ p ∈ E Γ ( p ) , ε = | E ( G ) | .

For a single graph, many SEMT labelings might exist and of course for a different labeling, there will be a different magic constant. For lower and upper bounds of the magic constants for subdivided stars, see [26,27]. Now we are concerned with evaluating the SEMT labeling and strength of imbalanced fork.

Theorem 2.3

For l ≥ 1 , the graph G ≅ Fr ( l , l + 1 , l + 2 ) is SEMT with magic constant:

a = 15 l + 22 2 ; l ≡ 0 ( mod 2 ) ; 5 ( 3 l + 5 ) 2 ; l ≡ 1 ( mod 2 ) .

Proof

Let G ≅ Fr ( l , l + 1 , l + 2 ) , l ≥ 1 and p = | V ( G ) | , q = | ( E ( G ) | , then p = 3 l + 4 , q = 3 l + 3 .

Consider the vertex labeling f : V ( G ) → { 1 , 2 , … , p } as follows:

For l ≡ 1 ( mod 2 ) :

f ( x 2 , 0 ) = l + 2 ,

f ( x ı , ȷ ) = 1 + ( l + 2 ) ı − 1 2 + ȷ − 1 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; l − ȷ − 2 2 + 1 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ; 3 l + 1 2 + l ı − 1 2 + ȷ − 2 2 + 4 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ; 3 l + 1 2 + l − ȷ − 1 2 + 3 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 .

From the above labeling “f”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = 6 + 3 ( l − 1 2 ) . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ + 1 , where ℏ + 1 = min ( S ) .

For l ≡ 0 ( mod 2 ) :

f ( x 2 , 0 ) = 2 l + 3 ,

f ( x ı , ȷ ) = 2 ( l + 1 ) + ( l + 2 ) ı − 1 2 − ȷ − 1 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; 2 ( l + 2 ) + ȷ − 2 2 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ; l 2 + ( l + 2 ) ı − 1 2 − ȷ − 2 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ; l 2 + ȷ − 1 2 + 1 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 .

From the above labeling “f”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q , where ℏ ′ = 6 + 3 ( l − 2 2 ) . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ ′ + 1 , where ℏ ′ + 1 = min ( S ) .□

This theorem gives us the magic constants a ( f ) = 15 l + 22 2 , for l ≡ 0 ( mod 2 ) , and a ( f ) = 5 ( 3 l + 5 ) 2 , for l ≡ 1 ( mod 2 ) of imbalanced fork, where l ≥ 1 and using the lower bound of magic constants in lemma obtained in [27], we have a ( f ) ≥ 5 q 2 + q + 12 2 q , where q = 3 l + 3 , thus we can conclude:

Theorem 2.4

The SEMT strength for Imbalanced Fork Fr ( l , l + 1 , l + 2 ) , l ≥ 1 (subdivision of star K 1 , 4 ) is

15 l 2 + 31 l + 20 2 ( l + 1 ) ≤ s m ( Fr ( l , l + 1 , l + 2 ) ) ≤ 15 l + 22 2 , l ≡ 0 ( mod 2 ) ,

15 l 2 + 31 l + 20 2 ( l + 1 ) ≤ s m ( Fr ( l , l + 1 , l + 2 ) ) ≤ 5 ( 3 l + 5 ) 2 , l ≡ 1 ( mod 2 ) .

2.1 SEMT labeling and deficiency of forests formed by imbalanced fork, star, bistar and path

In this section, it is shown that the forests consisting of imbalanced fork, star, bistar and path are SEMT with certain conditions on the parameters.

Theorem 2.6

For l ≥ 1 ,

Fr ( l , l + 1 , l + 2 ) ∪ K 1 , ϖ is SEMT,
μ s ( F r ( l , l + 1 , l + 2 ) ∪ K 1 , ϖ − 1 ) ≤ 1 ,

where

ϖ = l + 1 2 ; l ≡ 1 ( mod 2 ) ; l + 1 ; l ≡ 0 ( mod 2 ) .

Proof

Consider the graph G ≅ Fr ( l , l + 1 , l + 2 ) ∪ K 1 , ϖ .

Let p = | V ( G ) | and q = | E ( G ) | , then
p = 3 l + ϖ + 5 ,

q = 3 l + ϖ + 3 .

For l ≡ 1 ( mod 2 ) :

We define a labeling f : V ( Fr ( l , l + 1 , l + 2 ) ) → { 1 , 2 , … , 3 l + 4 } , as
f ( x ı , ȷ ) = l − ȷ − 1 2 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; 1 + l ı − 1 2 + ȷ − 2 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 .

Now consider the labeling g : V ( G ) → { 1 , 2 , … , p } .

For 1 ≤ k ≤ ϖ + 1 ,
g ( y k ) = 3 ( l + 1 ) 2 ; k = 1 ; 3 ( l + 1 ) + k ; k ≠ 1 .

Let A = 3 ( l + 1 ) 2 , then
f ( x ı , ȷ ) = A + ( l + 1 ) ı − 1 2 + ȷ − 1 2 + 1 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; A + l − ȷ − 2 2 + 1 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ,

f ( x 2 , 0 ) = g ( x 2 , 0 ) = 3 ( l + 1 ) + ϖ + 2 ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 .

From the above labeling “f”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = 3 ( l + 1 ) 2 + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ + 1 , where ℏ + 1 = min ( S ) .

For l ≡ 0 ( mod 2 ) :

We define a labeling f : V ( Fr ( l , l + 1 , l + 2 ) ) → { 1 , 2 , … , 3 l + 4 } , as
f ( x ı , ȷ ) = l + 2 2 + ȷ − 1 2 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; l 2 + ( l + 2 ) ı − 1 2 − ȷ − 2 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 .

Now consider the labeling g : V ( G ) → { 1 , 2 , … , p } .

For 1 ≤ k ≤ ϖ + 1 ,
g ( y k ) = 3 ( l + 2 ) 2 ; k = 1 ; 3 ( l + 1 ) + k ; k ≠ 1 .

Let A = 3 ( l + 2 ) 2 , then
f ( x ı , ȷ ) = A + l 2 + ( l + 1 ) ı − 1 2 − ȷ − 1 2 ; ı ≡ 1 ( mod 2 ) , ı = 1 , 3 ; ȷ ≡ 1 ( mod 2 ) , ȷ ≥ 1 ; A + l 2 + ȷ − 2 2 + 1 ; ı ≡ 0 ( mod 2 ) , ı = 2 ; ȷ ≡ 0 ( mod 2 ) , ȷ ≥ 2 ,

f ( x 2 , 0 ) = g ( x 2 , 0 ) = 3 ( l + 1 ) + ϖ + 2 ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 .

From the above labeling “f”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q , where ℏ ′ = 3 ( l + 2 ) 2 + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ ′ + 1 , where ℏ ′ + 1 = m i n ( S ) .
Let ℧ ≅ F r ( l , l + 1 , l + 2 ) ∪ K 1 , ϖ − 1 ∪ K 1

V ( ℧ ) = V ( F r ( l , l + 1 , l + 2 ) ) ∪ V ( K 1 , ϖ − 1 ) ∪ { z } .

Let p ′ = | V ( ℧ ) | and q ′ = | E ( ℧ ) | , then

p ′ = 3 l + ϖ + 5

and

q ′ = 3 l + ϖ + 2 .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in (a), we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2

with A = g ( y 1 ) = g ′ ( y 1 ) = 3 ( l + 1 ) 2 ,

g ′ ( y k ) = g ( y k ) ; 1 ≤ k ≤ ϖ ,

g ′ ( z ) = 3 ( l + 1 ) + ϖ + 1 ,

g ′ ( x 2 , 0 ) = g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + ϖ + 2 .

From the above labeling “ g ′ ”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = 3 ( l + 1 ) 2 + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + ℏ + 1 , where ℏ + 1 = min ( S ) .

For l ≡ 0 ( mod 2 ) :

Keeping in mind the valuation f defined in (a), we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2

with A = g ( y 1 ) = g ′ ( y 1 ) = 3 ( l + 2 ) 2 ,

g ′ ( y k ) = g ( y k ) ; 1 ≤ k ≤ ϖ ,

g ′ ( z ) = 3 ( l + 1 ) + ϖ + 1 ,

g ′ ( x 2 , 0 ) = g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + ϖ + 2 .

From the above labeling “ g ′ ”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q ′ , where ℏ ′ = 3 ( l + 2 ) 2 + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + ℏ ′ + 1 , where ℏ ′ + 1 = min ( S ) .

Theorem 2.7

For l ≥ 1 ,

Fr ( l , l + 1 , l + 2 ) ∪ B S ( ζ , ξ ) is SEMT,
μ s ( Fr ( l , l + 1 , l + 2 ) ∪ B S ( ζ , ξ − 1 ) ) ≤ 1 , l ≠ 1 ,

where ζ ≥ 0 and

ξ = l − 1 2 ; l ≡ 1 ( mod 2 ) ; l ; l ≡ 0 ( mod 2 ) .

Proof

Consider the graph G ≅ Fr ( l , l + 1 , l + 2 ) ∪ B S ( ζ , ξ ) .

Let p = | V ( G ) | and q = | E ( G ) | , then
p = 3 l + ζ + ξ + 6 ,

q = 3 l + ζ + ξ + 4 .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6 with A = 3 l + 1 2 + ζ + 1 , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as
g ( z ♭ t ) = 3 l + 1 2 + t ; ♭ = 1 , 1 ≤ t ≤ ζ ; 3 l + 1 2 + ζ + 1 ; ♭ = 2 , t = 0 ; 3 ( l + 1 ) + ζ + 2 ; ♭ = 1 , t = 0 ; 3 ( l + 1 ) + ζ + t + 2 ; ♭ = 2 , 1 ≤ t ≤ ξ ,

f ( x ı , ȷ ) = g ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 ,

f ( x 2 , 0 ) = g ( x 2 , 0 ) = 3 ( l + 1 ) + ζ + ξ + 3 .

From the above labeling “g”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = 3 l + 1 2 + ζ + 2 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ + 1 , where ℏ + 1 = min ( S ) .

For l ≡ 0 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6 with A = 3 l + 4 2 + ζ + 1 , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as
g ( z ♭ t ) = 3 l + 4 2 + t ; ♭ = 1 , 1 ≤ t ≤ ζ ; 3 l + 4 2 + ζ + 1 ; ♭ = 2 , t = 0 ; 3 ( l + 1 ) + ζ + 2 ; ♭ = 1 , t = 0 ; 3 ( l + 1 ) + ζ + t + 2 ; ♭ = 2 , 1 ≤ t ≤ ξ ,

f ( x ı , ȷ ) = g ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 ,

f ( x 2 , 0 ) = g ( x 2 , 0 ) = 3 ( l + 1 ) + ζ + ξ + 3 .

From the above labeling “g”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q , where ℏ ′ = 3 l + 4 2 + ζ + 2 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + ℏ ′ + 1 , where ℏ ′ + 1 = min ( S ) .
Let ℧ ≅ Fr ( l , l + 1 , l + 2 ) ∪ BS ( ζ , ξ − 1 ) ∪ K 1 .

Here,

V ( ℧ ) = V ( Fr ( l , l + 1 , l + 2 ) ) ∪ V ( BS ( ζ , ξ − 1 ) ) ∪ { z } .

Let p ′ = | V ( ℧ ) | and q ′ = | E ( ℧ ) | , then

p ′ = 3 l + ζ + ξ + 6

and

q ′ = 3 l + ζ + ξ + 3 .

For l ≡ 1 ( mod 2 ) :

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2

with A = g ( z 20 ) = g ′ ( z 20 ) = 3 l + 1 2 + ζ + 1 ,

g ′ ( z 1 t ) = g ( z 1 t ) ; 0 ≤ t ≤ ζ ,

g ′ ( z 2 t ) = g ( z 2 t ) ; 0 ≤ t ≤ ξ − 1 ,

g ′ ( z ) = 3 ( l + 1 ) + ζ + ξ + 2 ,

g ′ ( x 2 , 0 ) = g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + ζ + ξ + 3 .

From the above labeling “ g ”, we obtain consecutive Z + from ℏ + 1 to ℏ + q ′ , where ℏ = 3 l + 1 2 + ζ + 2 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + ℏ + 1 , where ℏ + 1 = min ( S ) .

For l ≡ 0 ( mod 2 ) :

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2

with A = g ( z 20 ) = g ′ ( z 20 ) = 3 l + 4 2 + ζ + 1 ,

g ′ ( z 1 t ) = g ( z 1 t ) ; 0 ≤ t ≤ ζ ,

g ′ ( z 2 t ) = g ( z 2 t ) ; 0 ≤ t ≤ ξ − 1 ,

g ′ ( z ) = 3 ( l + 1 ) + ζ + ξ + 2 ,

g ′ ( x 2 , 0 ) = g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + ζ + ξ + 3 .

From the above labeling “ g ”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q ′ , where ℏ ′ = 3 l + 4 2 + ζ + 2 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + ℏ ′ + 1 , where ℏ ′ + 1 = min ( S ) .□

In the following theorems, we present the two distinct SEMT labelings – which are non-dual of each other – for the same forest be composed of disjoint union of path P m and imbalanced fork.

Theorem 2.8

For l ≥ 1 ,

(a)(i): Fr ( l , l + 1 , l + 2 ) ∪ P r is SEMT,
(a)(ii): Fr ( l , l + 1 , l + 2 ) ∪ P r − 1 is SEMT,
(b)(i): μ s ( Fr ( l , l + 1 , l + 2 ) ∪ P r − 2 ) ≤ 1 ,
(b)(ii): μ s ( Fr ( l , l + 1 , l + 2 ) ∪ P r − 3 ) ≤ 1 ,

where

r = l + 2 ; l ≡ 1 ( mod 2 ) ; 2 l + 3 ; l ≡ 0 ( mod 2 ) .

Proof

Consider the graph G ≅ Fr ( l , l + 1 , l + 2 ) ∪ P ϱ , where

ϱ = r ; for a ( i ) ; r − 1 ; for a ( i i ) .

Let p = | V ( G ) | and q = | E ( G ) | , so we get

p = 3 l + ϱ + 4 ,

q = 3 l + ϱ + 2 .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6 with A = ⌈ 3 l 2 ⌉ + ⌊ ϱ + 1 2 ⌋ , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as

g ( x t ) = 3 l 2 + k ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 ; 9 ( l + 1 ) 2 + k − l ; t = 2 k , 1 ≤ k ≤ ϱ 2 , for a ( i ) ; 9 ( l + 1 ) 2 + k − l − 1 ; t = 2 k , 1 ≤ k ≤ ϱ 2 , for a ( i i ) ,

g ( x 2 , 0 ) = f ( x 2 , 0 ) = 9 ( l + 1 ) 2 + ϱ 2 − l + 1 ; for a ( i ) ; 9 ( l + 1 ) 2 + ϱ 2 − l ; for a ( i i ) ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 .

From the above labeling “g”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = ⌈ 3 l 2 ⌉ + ⌊ ϱ + 1 2 ⌋ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + min ( S ) , where min ( S ) = ℏ + 1 .

For l ≡ 0 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6 with A = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌊ ϱ + 1 2 ⌋ , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as

g ( x t ) = 3 ( l + 1 ) 2 + k ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 + 1 ; t = 2 k , 1 ≤ k ≤ ϱ 2 , for a ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 ; t = 2 k , 1 ≤ k ≤ ϱ 2 , for a ( i i ) ,

g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + 3 ( l + 1 ) 2 + ϱ 2 − l 2 + 2 ; for a ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + ϱ 2 − l 2 + 1 ; for a ( i i ) ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 .

From the above labeling “g”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q , where ℏ ′ = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌊ ϱ + 1 2 ⌋ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + min ( S ) , where min ( S ) = ℏ ′ + 1 .

Let ℧ ≅ F r ( l , l + 1 , l + 2 ) ∪ P ϱ ∪ K 1 , where

V ( ℧ ) = V ( F r ( l , l + 1 , l + 2 ) ) ∪ V ( P ϱ ) ∪ { z } .

Let p ′ = | V ( ℧ ) | and q ′ = | E ( ℧ ) | , so we get

p ′ = 3 l + ϱ + 5 ,

q ′ = 3 l + ϱ + 2 ,

where

ϱ = r − 2 ; for b ( i ) ; r − 3 ; for b ( i i ) .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6, we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both b ( i ) and b ( i i )

with A = ⌈ 3 l 2 ⌉ + ⌊ ϱ + 1 2 ⌋

g ′ ( x t ) = g ( x t ) , t ≡ 1 ( mod 2 ) ,

g ′ ( x t ) = 9 ( l + 1 ) 2 + k − l − 1 ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 , for b ( i ) ; 9 ( l + 1 ) 2 + k − l − 2 ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 , for b ( i i ) .

Let B = 9 ( l + 1 ) 2 + ϱ − 1 2 − l − 1 and C = 9 ( l + 1 ) 2 + ⌈ ϱ − 1 2 ⌉ − l − 2 , then

g ′ ( z ) = B + 1 ; for b ( i ) ; C + 1 ; for b ( i i ) ,

g ′ ( x 2 , 0 ) = B + 2 ; for b ( i ) ; C + 2 ; for b ( i i ) .

From the above labeling “ g ′ ”, we obtain consecutive Z + from ℏ + 1 to ℏ + q ′ , where ℏ = ⌈ 3 l 2 ⌉ + ⌊ ϱ + 1 2 ⌋ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + min ( S ) , where min ( S ) = ℏ + 1 .

For l ≡ 0 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6, we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both b ( i ) and b ( i i )

with A = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌊ ϱ + 1 2 ⌋ ,

g ′ ( x t ) = g ( x t ) , t ≡ 1 ( mod 2 ) ,

g ′ ( x t ) = 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 , for b ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 − 1 ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 , for b ( i i ) .

Let B = 3 ( l + 1 ) + ⌊ 3 ( l + 1 ) 2 ⌋ + ϱ − 1 2 − l 2 and C = 3 ( l + 1 ) + ⌊ 3 ( l + 1 ) 2 ⌋ + ⌈ ϱ − 1 2 ⌉ − l 2 − 1 , then

g ′ ( z ) = B + 1 ; for b ( i ) ; C + 1 ; for b ( i i ) ,

g ′ ( x 2 , 0 ) = B + 2 ; for b ( i ) ; C + 2 ; for b ( i i ) .

From the above labeling “ g ”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q ′ , where ℏ ′ = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌊ ϱ + 1 2 ⌋ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + min ( S ) , where min ( S ) = ℏ ′ + 1 .□

Theorem 2.9

For l ≥ 1 ,

(a)(i): Fr ( l , l + 1 , l + 2 ) ∪ P r is SEMT,
(a)(ii): Fr ( l , l + 1 , l + 2 ) ∪ P r − 1 is SEMT, l ≠ 1 ,
(b)(i): μ s ( Fr ( l , l + 1 , l + 2 ) ∪ P r − 2 ) ≤ 1 ,
(b)(ii): μ s ( Fr ( l , l + 1 , l + 2 ) ∪ P r − 3 ) ≤ 1 , l ≠ 1 ,

where

r = l + 1 ; l ≡ 1 ( mod 2 ) ; 2 ( l + 1 ) ; l ≡ 0 ( mod 2 ) .

Proof

Consider the graph G ≅ F r ( l , l + 1 , l + 2 ) ∪ P ϱ , where
ϱ = r ; for a ( i ) ; r − 1 ; for a ( i i ) .

Let p = | V ( G ) | and q = | E ( G ) | , so we get
p = 3 l + ϱ + 4 ,

q = 3 l + ϱ + 2 .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6 with A = ⌈ 3 l 2 ⌉ + ⌈ ϱ − 1 2 ⌉ , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as
g ( x t ) = 3 l 2 + k ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 ; 9 ( l + 1 ) 2 − l + k − 1 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ 2 , for a ( i ) ; 9 ( l + 1 ) 2 − l + k − 2 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ 2 , for a ( i i ) ,

g ( x 2 , 0 ) = f ( x 2 , 0 ) = 9 ( l + 1 ) 2 − l + ϱ 2 ; for a ( i ) ; 9 ( l + 1 ) 2 − l + ϱ 2 − 1 ; for a ( i i ) ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both a ( i ) and a ( i i ) .

From the above labeling “g”, we obtain consecutive Z + from ℏ + 1 to ℏ + q , where ℏ = 3 l 2 + ϱ − 1 2 + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + min ( S ) , where min ( S ) = ℏ + 1 .

For l ≡ 0 (mod 2):

Keeping in mind the valuation f defined in Theorem 2.6 with A = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌈ ϱ − 1 2 ⌉ , we describe the labeling g : V ( G ) → { 1 , 2 , … , p } as
g ( x t ) = 3 ( l + 1 ) 2 + k ; t = 2 k , 1 ≤ k ≤ ϱ − 1 2 ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ 2 , for a ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 − 1 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ 2 , for a ( i i ) ,

g ( x 2 , 0 ) = f ( x 2 , 0 ) = 3 ( l + 1 ) + 3 ( l + 1 ) 2 + ϱ 2 − l 2 + 1 ; for a ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + ϱ 2 − l 2 ; for a ( i i ) ,

g ( x ı , ȷ ) = f ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both a ( i ) and a ( i i ) .

From the above labeling “g′”, we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q , where ℏ ′ = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌈ ϱ − 1 2 ⌉ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p + q + min ( S ) , where min ( S ) = ℏ ′ + 1 .
Let ℧ ≅ F r ( l , l + 1 , l + 2 ) ∪ P ϱ ∪ K 1 , where

V ( ℧ ) = V ( F r ( l , l + 1 , l + 2 ) ) ∪ V ( P ϱ ) ∪ { z } ,

where

ϱ = r − 2 ; for b ( i ) ; r − 3 ; for b ( i i ) .

For l ≡ 1 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6, we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both b ( i ) and b ( i i )

with A = ⌈ 3 l 2 ⌉ + ⌈ ϱ − 1 2 ⌉ ,

g ′ ( x t ) = g ( x t ) , t ≡ 0 ( mod 2 ) ,

g ′ ( x t ) = 9 ( l + 1 ) 2 + k − l − 2 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 , for b ( i ) ; 9 ( l + 1 ) 2 + k − l − 3 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 , for b ( i i ) .

Let B = 9 ( l + 1 ) 2 + ⌊ ϱ + 1 2 ⌋ − l − 2 and C = 9 ( l + 1 ) 2 + ϱ + 1 2 − l − 3 , then

g ′ ( z ) = B + 1 ; for b ( i ) ; C + 1 ; for b ( i i ) ,

g ′ ( x 2 , 0 ) = B + 2 ; for b ( i ) ; C + 2 ; for b ( i i ) .

From the above labeling “ g ′ ”, we obtain consecutive Z + from ℏ + 1 to ℏ + q ′ , where ℏ = ⌈ 3 l 2 ⌉ + ⌈ ϱ − 1 2 ⌉ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + min ( S ) , where min ( S ) = ℏ + 1 .

For l ≡ 0 ( mod 2 ) :

Keeping in mind the valuation f defined in Theorem 2.6, we describe the labeling g ′ : V ( ℧ ) → { 1 , 2 , … , p ′ } as

f ( x ı , ȷ ) = g ( x ı , ȷ ) = g ′ ( x ı , ȷ ) ; 1 ≤ ı ≤ 3 , 1 ≤ ȷ ≤ ε , l ≤ ε ≤ l + 2 , for both b ( i ) and b ( i i )

with A = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌈ ϱ − 1 2 ⌉

g ′ ( x t ) = g ( x t ) , t ≡ 0 ( mod 2 ) ,

g ′ ( x t ) = 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 − 1 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 , for b ( i ) ; 3 ( l + 1 ) + 3 ( l + 1 ) 2 + k − l 2 − 2 ; t = 2 k − 1 , 1 ≤ k ≤ ϱ + 1 2 , for b ( i i ) .

Let B = 3 ( l + 1 ) + ⌊ 3 ( l + 1 ) 2 ⌋ + ⌊ ϱ + 1 2 ⌋ − l 2 − 1 and C = 3 ( l + 1 ) + ⌊ 3 ( l + 1 ) 2 ⌋ + ϱ + 1 2 − l 2 − 2 , then

g ′ ( z ) = B + 1 ; for b ( i ) ; C + 1 ; for b ( i i ) ,

g ′ ( x 2 , 0 ) = B + 2 ; for b ( i ) ; C + 2 ; for b ( i i ) .

From the above labeling “ g ′ ,” we obtain consecutive Z + from ℏ ′ + 1 to ℏ ′ + q ′ , where ℏ ′ = ⌈ 3 ( l + 1 ) 2 ⌉ + ⌈ ϱ − 1 2 ⌉ + 1 . Thus, Lemma 2.2 gives the required result with magic constant a = p ′ + q ′ + min ( S ) , where min ( S ) = ℏ ′ + 1 .□

References

[1] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970), 451–461.10.4153/CMB-1970-084-1Search in Google Scholar

[2] H. Enomoto, A. S. Lladó, T. Nakamigawa, and G. Ringel, Super edge-magic graphs, SUT J. Math. 34 (1998), no. 2, 105–109.10.55937/sut/991985322Search in Google Scholar

[3] S. Avadayappan, P. Jeyanthi, and R. Vasuki, Super magic strength of a graph, Indian J. Pure Appl. Math. 32 (2001), no. 11, 1621–1630.Search in Google Scholar

[4] R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, On the super edge-magic deficiency of graphs, Ars Combin. 78 (2006), 33–45.10.1016/S1571-0653(04)00074-5Search in Google Scholar

[5] R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, On the super edge-magic deficiency of graphs, Electron. Notes Discrete Math. 11 (2002), 299–314.10.1016/S1571-0653(04)00074-5Search in Google Scholar

[6] R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, Some new results on the super edge-magic deficiency of graphs, J. Combin. Math. Combin. Comput. 55 (2005), 17–31.Search in Google Scholar

[7] A. A. Ngurah, E. T. Baskoro, and R. Simanjuntak, On the super edge-magic deficiencies of graphs, Australas. J. Combin. 40 (2008), 3–14.Search in Google Scholar

[8] S. Javed, A. Riasat, and S. Kanwal, On super edge magicness and deficiencies of forests, Utilitas Math. 98 (2015), 149–169.Search in Google Scholar

[9] K. Ali, M. Hussain, H. Shaker, and M. Javed, Super edge-magic total labeling of subdivided stars, Ars Combin. 120 (2015), 161–167.Search in Google Scholar

[10] V. Swamminatan and P. Jeyanthi, Super edge-magic strength of fire crackers, banana trees and unicyclic graphs, Discrete Math. 306 (2006), 1624–1636.10.1016/j.disc.2005.06.038Search in Google Scholar

[11] D. G. Akka and N. S. Warad, Super magic strength of a graph, Indian J. Pure Appl. Math. 41 (2010), no. 4, 557–568.10.1007/s13226-010-0031-zSearch in Google Scholar

[12] S. Kanwal, A. Azam, and Z. Iftikhar, SEMT labelings and deficiencies of forests with two components (II), Punjab Univ. J. Math. 51 (2019), no. 4, 1–12.Search in Google Scholar

[13] S. Kanwal, Z. Iftikhar, and A. Azam, SEMT labelings and deficiencies of forests with two components (I), Punjab Univ. J. Math. 51 (2019), no. 5, 137–149.Search in Google Scholar

[14] S. Kanwal, M. Imtiaz, Z. Iftikhar, R. Ashraf, M. Arshad, R. Irfan, and T. Sumbal, Embedding of supplementary results in strong EMT valuations and strength, Open Math. 17 (2019), no. 1, 527–543.10.1515/math-2019-0044Search in Google Scholar

[15] S. Kanwal and I. Kanwal, SEMT valuations of disjoint union of combs, stars and banana trees, Punjab Univ. J. Math. 50 (2018), no. 3, 131–144.Search in Google Scholar

[16] N. S. Hungund and D. G. Akka, Super edge-magic strength of some new families of graphs, Bull. Marathwada Math. Soc. 12 (2011), no. 1, 47–54.Search in Google Scholar

[17] M. Javaid and A. A. Bhatti, On super (a,d)-edge-antimagic total labeling of subdivided stars, Ars Combin. 105 (2012), 503–512.Search in Google Scholar

[18] M. Javed, M. Hassain, K. Ali, and H. Shaker, On super edge-magic total labeling on subdivision of trees, Util. Math. 89 (2012), 169–177.Search in Google Scholar

[19] A. Ali, M. Javaid, and M. A. Rehman, SEMT labeling on disjoint union of subdivided stars, Punjab Uni. J. Math. 48 (2016), no. 1, 111–122.Search in Google Scholar

[20] V. Swaminathan and P. Jeyanthi, Super edge-magic strength of generalised theta graphs, Int. J. Inf. Manag. Sci. 22 (2006), no. 3, 203–220.Search in Google Scholar

[21] V. Swaminathan and P. Jeyanthi, Strong super edge-magic graphs, Math. Educ. XLII (2008), no. 3, 156–160.Search in Google Scholar

[22] V. Swaminathan and P. Jeyanthi, Super edge-magic labeling of some new classes of graphs, Math. Educ. XLII (2008), no. 2, 91–94.Search in Google Scholar

[23] J. A. Gallian, A dynamic survey of graph labeling, 22th edition, Electron. J. Combin. (Dec. 2019), # DS6.Search in Google Scholar

[24] S. Kanwal, A. Riasat, M. Imtiaz, Z. Iftikhar, S. Javed, and R. Ashraf, Bounds of strong EMT strength for certain subdivision of star and bistar, Open Math. 16 (2018), 1313–1325.10.1515/math-2018-0111Search in Google Scholar

[25] R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, The place of super edge-magic labeling among other classes of labeling, Discrete Math. 231 (2001), 153–168.10.1016/S0012-365X(00)00314-9Search in Google Scholar

[26] E. T. Baskoro, A. A. G. Ngurah, and R. Simanjuntak, On (super) edge-magic total labeling of subdivision of K1,3, SUT J. Math. 43 (2007), 127–136.10.55937/sut/1252506095Search in Google Scholar

[27] M. Javaid, Labeling Graphs and Hypergraphs, PhD thesis, FAST-NUCES, Lahore Campus, Pakistan, 2013.Search in Google Scholar

Received: 2019-03-26

Revised: 2020-06-02

Accepted: 2020-07-31

Published Online: 2020-10-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

SEMT valuation and strength of subdivided star of K _1,4

Abstract

1 Preliminaries

2 Main results

Definition 2.1

Lemma 2.2

Theorem 2.3

Proof

Theorem 2.4

2.1 SEMT labeling and deficiency of forests formed by imbalanced fork, star, bistar and path

Theorem 2.6

Proof

Theorem 2.7

Proof

Theorem 2.8

Proof

Theorem 2.9

Proof

References

Journal and Issue

Articles in the same Issue

SEMT valuation and strength of subdivided star of K 1,4

Abstract

1 Preliminaries

2 Main results

Definition 2.1

Lemma 2.2

Theorem 2.3

Proof

Theorem 2.4

2.1 SEMT labeling and deficiency of forests formed by imbalanced fork, star, bistar and path

Theorem 2.6

Proof

Theorem 2.7

Proof

Theorem 2.8

Proof

Theorem 2.9

Proof

References

Journal and Issue

Articles in the same Issue

SEMT valuation and strength of subdivided star of K _1,4