Embedding of Supplementary Results in Strong EMT Valuations and Strength

Theorem 1

The SEMT strength for 2-sided generalized comb Cbν,υ2 ; ν ≥ 2, υ be an odd number ≥ 3 is, for q = νυ:

Definition 2

Star on n vertices is denoted by K_1,n–1, is given by the following set of vertices and edges:

Bistar BS(ζ, ξ) is an acyclic graph on n vertices obtained from two stars K_1,ζ and K_1,ξ by joining their central vertices by an edge, where ζ, ξ ≥ 1 and ζ + ξ = n – 2,

where

Definition 3

[25] A caterpillar is a graph derived from a path by hanging any number of leaves from the vertices of the path. The caterpillar can be seen as a sequence of stars S₁ ∪ S₂ ∪ … ∪ S_m, where each S_ı is a star with central vertex c_ı and η_ı leaves for ı = 1, 2, …, m and the leaves of S_ı include c_ı–1 and c_ı+1, for ı = 2, 3, …, m–1. We denoted the caterpillar as S_{η1,η2,…,ηm}, where the vertex and edge sets are as respectively, V(S_{η1,η2,…,ηm}) = {c_ı : 1 ≤ ı ≤ m} ∪ ⋃ı=2m−1{υıȷ:2≤ȷ≤ηı−1}∪{υ1ȷ:1≤ȷ≤η1−1}∪{υmȷ:2≤ȷ≤ηm}, E(S_{η1,η2,…,ηm}) = {c_ıc_ı+1 : 1 ≤ ı ≤ m – 1} ∪ ⋃ı=2m−1{cıυıȷ:2≤ȷ≤ηı−1}∪{c1υ1ȷ:1≤ȷ≤η1−1}∪{cmυmȷ:2≤ȷ≤ηm} and |V(Sη1,η2,…,ηm)|=∑ı=1mηı−m+2,|E(Sη1,η2,…,ηm)|=∑ı=1mηı−m+1.

Theorem 2

For ν ≥ 2, υ = 2ε + 1, ε = 1, 2, 3, …

1. Cbν,υ2 ∪ BS(ζ, ξ) is SEMT.

2. μ_s( Cbν,υ2 ∪ BS(ζ, ξ – 1)) ≤ 1.

   where ζ ≥ 0 and

   ξ=1+υ(ν2−1)+1⌊υ−14⌋+2⌊υ−24⌋  ; ν≡0(mod2)2+υ(ν−32)+3⌊υ−14⌋+2⌊υ−24⌋  ; ν≡1(mod2)

Proof

1. Consider the graph ℘ ≅ Cbν,υ2 ∪ BS(ζ, ξ)

   Let p = |V(℘)| and q = |E(℘)|, then we get

   p=νυ+ζ+ξ+3q=νυ+ζ+ξ+1

   we define a labeling Υ : V( Cbν,υ2 ) → {1, 2, …, νυ + 1} as

   Υ(xυ+12,0)=νυ+ζ+ξ+3

   and consider the labeling Ψ : V(℘) → {1, 2, …, p}.

   For υ−12 odd;

   Υ(xı,ȷ)=ı2+υ(ȷ−1)2 ;ı≡0(mod2),ȷ≡1(mod2)υȷ2−ı−12 ;ı≡1(mod2),ȷ≡0(mod2)

   Ψ(z♭t)=⌈νυ2⌉+t ;♭=1,1≤t≤ζ⌈νυ2⌉+ζ+1 ;♭=2,t=0

   Let A = ⌊νυ2⌋ + ζ + 1, then

   Υ(xı,ȷ)=A+ı+12+υ(ȷ−1)2 ;ı,ȷ≡1(mod2)A+υȷ2−ı2+1  ;ı,ȷ≡0(mod2)

   

   Figure 3

   SEMT forest Cb4,92 ∪ BS(5, 14)

   Ψ(z♭t)=νυ+ζ+2   ;♭=1,t=0νυ+ζ+2+t ;♭=2,1≤t≤ξ

   so,

   Υ(xυ+12,0)=Ψ(xυ+12,0)

   and

   Υ(xı,ȷ)=Ψ(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ + 1, ħ + 2, …, ħ + q}, where ħ = ⌊νυ2⌋ + ζ + 2. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + ħ + 1, where ħ + 1 = min(S).

   For υ−12 even;

   Υ(xı,ȷ)=ı+12+υ(ȷ−1)2 ;ı,ȷ≡1(mod2)υȷ2−ı2+1  ;ı,ȷ≡0(mod2)

   Ψ(z♭t)=⌈νυ2⌉+t ;♭=1,1≤t≤ζ⌈νυ2⌉+ζ+1 ;♭=2,t=0

   Let A′ = ⌈νυ2⌉ + ζ + 1, then

   Υ(xı,ȷ)=A′+υ(ȷ−1)2+ı2 ;ı≡0(mod2),ȷ≡1(mod2)A′+υȷ2−ı−12 ;ı≡1(mod2),ȷ≡0(mod2)

   Ψ(z♭t)=νυ+ζ+2 ;♭=1,t=0νυ+ζ+2+t ;♭=2,1≤t≤ξ

   so,

   Υ(xυ+12,0)=Ψ(xυ+12,0)

   and

   Υ(xı,ȷ)=Ψ(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ′ + 1, ħ′ + 2, …, ħ′ + q}, where ħ′ = ⌈νυ2⌉ + ζ + 2. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + ħ′ + 1, where ħ′ + 1 = min(S).

2. Let ℧ ≅ Cbν,υ2 ∪ BS(ζ, ξ – 1) ∪ K₁

   Here

   V(℧)=V(Cbν,υ2)∪V(BS(ζ,ξ−1))∪{z}

   and

   V(BS(ζ,ξ−1)={z♭t:♭=1,2;0≤t≤Λ},

   where

   Λ=ζ;♭=1ξ−1;♭=2

   E(BS(ζ,ξ−1))={z10z1t;1≤t≤ζ}∪{z10z20}∪{z20z2t;1≤t≤ξ−1}

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

   p′=νυ+ζ+ξ+3q′=νυ+ζ+ξ

   Before formulating the labeling Ψ′ : V(℧) → {1, 2, …, p′}, keep in view the labeling Υ defined in (a). We define the labeling Ψ′ as follows:

   For υ−12 odd;

   Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   with A = Ψ(z₂₀) = Ψ′(z₂₀)

   Ψ′(z1t)=Ψ(z1t),0≤t≤ζΨ′(z2t)=Ψ(z2t),0≤t≤ξ−1Ψ′(z)=νυ+ζ+ξ+2Ψ′(xυ+12,0)=Ψ(xυ+12,0)=Υ(xυ+12,0)

   The edge-sums generated by the above labeling ”Ψ′” are the set of consecutive positive integers S = {ħ + 1, ħ + 2, …, ħ + q′}, where ħ = ⌊νυ2⌋ + ζ + 2. Thus by Lemma 1, ”Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + ħ + 1, where ħ + 1 = min(S).

   For υ−12 even;

   Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   with A′ = Ψ(z₂₀) = Ψ′(z₂₀)

   Ψ′(z1t)=Ψ(z1t);0≤t≤ζΨ′(z2t)=Ψ(z2t);0≤t≤ξ−1Ψ′(z)=νυ+ζ+ξ+2Ψ′(xυ+12,0)=Ψ(xυ+12,0)=Υ(xυ+12,0)

   The edge-sums generated by the above labeling ”Ψ′” are the set of consecutive positive integers S = {ħ′ + 1, ħ′ + 2, …, ħ′ + q′}, where ħ′ = ⌈νυ2⌉ + ζ + 2. Thus by Lemma 1, ”Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + ħ′ + 1, where ħ′ + 1 = min(S).□

Theorem 3

For ν ≥ 2, υ = 2ε + 1, ε = 1, 2, 3, …

1. Cbν,υ2 ∪ S_ζ,ξ,η is SEMT.

2. μ_s( Cbν,υ2 ∪ S_ζ,ξ–1,η) ≤ 1; (ν, υ) ≠ (2, 3)

   where ζ, η ≥ 2 and

   ξ=2+υ(ν2−1)+1⌊υ−14⌋+2⌊υ−24⌋  ;ν≡0(mod2)3+υ(ν−32)+3⌊υ−14⌋+2⌊υ−24⌋  ;ν≡1(mod2)

Proof

1. Consider the graph ℘ ≅ Cbν,υ2 ∪ S_ζ,ξ,η, where

   V(Sζ,ξ,η)={cı:1≤ı≤3}∪{xı;1≤ı≤ζ−1}∪{yı;2≤ı≤ξ−1}∪{zı;2≤ı≤η}

   and

   E(Sζ,ξ,η)={cıcı+1:1≤ı≤2}∪{c1xı;1≤ı≤ζ−1}∪{c2yı;2≤ı≤ξ−1}∪{c3zı;2≤ı≤η}

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

   p=νυ+ζ+ξ+ηq=νυ+ζ+ξ+η−2

   Before formulating the labeling Ψ : V(℘) → {1, 2, …, p}, keep in view the labeling Υ defined in Theorem 2.

   For υ−12 odd;

   Take the previously mentioned labeling Υ (for υ−12 odd) with A = ⌊νυ2⌋ + ζ + η – 1. We define the labeling Ψ as follows:

   Ψ(xı)=⌊νυ2⌋+ı;1≤ı≤ζ−1Ψ(c2)=⌊νυ2⌋+ζΨ(zı)=⌊νυ2⌋+ζ+ı;1≤ı≤η−1Ψ(c1)=νυ+ζ+ηΨ(yı)=νυ+ζ+η+ı;1≤ı≤ξ−2Ψ(c3)=νυ+ζ+ξ+η−1Ψ(xυ+12,0)=νυ+ζ+ξ+η

   Ψ(x_ı,ȷ) = Υ(x_ı,ȷ) ; 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ + 1, ħ + 2, …, ħ + q}, where ħ = ⌊νυ2⌋ + ζ + η. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + ħ + 1, where ħ + 1 = min(S).

   For υ−12 even;

   Take the previously mentioned labeling Υ (for υ−12 odd) with A′ = ⌈νυ2⌉ + ζ + η – 1. We define the labeling Ψ as follows:

   Ψ(xı)=⌈νυ2⌉+ı;1≤ı≤ζ−1Ψ(c2)=⌈νυ2⌉+ζΨ(zı)=⌈νυ2⌉+ζ+ı;1≤ı≤η−1

   

   Figure 4

   SEMT forest Cb5,92 ∪ S_6,19,11 ∪ K₁

   Ψ(c1)=νυ+ζ+ηΨ(yı)=νυ+ζ+η+ı;1≤ı≤ξ−2Ψ(c3)=νυ+ζ+ξ+η−1Ψ(xυ+12,0)=νυ+ζ+ξ+η

   Ψ(x_ı,ȷ) = Υ(x_ı,ȷ) ; 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ′ + 1, ħ′ + 2, …, ħ′ + q}, where ħ′ = ⌈νυ2⌉ + ζ + η. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + ħ′ + 1, where ħ′ + 1 = min(S).

2. Let ℧ ≅ Cbν,υ2 ∪ S_ζ,ξ–1,η ∪ K₁, where

   V(℧)=V(Cbν,υ2)∪V(Sζ,ξ−1,η)∪{d}V(Sζ,ξ−1,η)={cı:1≤ı≤3}∪{xı;1≤ı≤ζ−1}∪{yı;2≤ı≤ξ−2}∪{zı;2≤ı≤η}

   and

   E(Sζ,ξ−1,η)={cıcı+1:1≤ı≤2}∪{c1xı;1≤ı≤ζ−1}∪{c2yı;2≤ı≤ξ−2}∪{c3zı;2≤ı≤η}

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

   p′=νυ+ζ+ξ+ηq′=νυ+ζ+ξ+η−3

   Before formulating the labeling Ψ′ : V(℧) → {1, 2, …, p′}, keep in view the labeling Υ defined in Theorem 2.

   For υ−12 odd;

   Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   with A = ⌊νυ2⌋ + ζ + η – 1

   Ψ′(xı)=Ψ(xı);1≤ı≤ζ−1Ψ′(c2)=Ψ(c2)Ψ′(zı)=Ψ(zı);1≤ı≤η−1Ψ′(c1)=Ψ(c1)Ψ′(yı)=νυ+ζ+η+ı;1≤ı≤ξ−3Ψ′(c3)=νυ+ζ+ξ+η−2Ψ′(d)=νυ+ζ+ξ+η−1Ψ′(xυ+12,0)=Ψ(xυ+12,0)

   The edge-sums generated by the above labeling ”Ψ′” are the set of consecutive positive integers S = {ħ + 1, ħ + 2, …, ħ + q′}, where ħ = ⌊νυ2⌋ + ζ + η. Thus by Lemma 1, ”Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + ħ + 1, where ħ + 1 = min(S).

   For υ−12 even;

   Υ(xı,ȷ)=Ψ(xı,ȷ)=Ψ′(xı,ȷ);1≤ı≤υ,1≤ȷ≤ν

   with A′ = ⌈νυ2⌉ + ζ + η – 1

   Ψ′(xı)=Ψ(xı);1≤ı≤ζ−1Ψ′(c2)=Ψ(c2)Ψ′(zı)=Ψ(zı);1≤ı≤η−1Ψ′(c1)=Ψ(c1)Ψ′(yı)=νυ+ζ+η+ı;1≤ı≤ξ−3Ψ′(c3)=νυ+ζ+ξ+η−2Ψ′(d)=νυ+ζ+ξ+η−1Ψ′(xυ+12,0)=Ψ(xυ+12,0)

   The edge-sums generated by the above labeling ”Ψ′” are the set of consecutive positive integers S = {ħ′ + 1, ħ′ + 2, …, ħ′ + q′}, where ħ′ = ⌈νυ2⌉ + ζ + η. Thus by Lemma 1, ”Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + ħ′ + 1, where ħ′ + 1 = min(S).□

Definition 4

P_n be a path of order n and length n – 1, with vertices labelled from ν₁ to ν_n along P_n and E(P_n) = {ν_ıν_ı+1; 1 ≤ ı ≤ n – 1}.

Theorem 4

For ν ≥ 2, υ = 2ε + 1, ε = 1, 2, 3, …

1. Cbν,υ2 ∪ P_r is SEMT.

2. Cbν,υ2 ∪ P_r–1 is SEMT.

3. μ_s( Cbν,υ2 ∪ P_r–2) ≤ 1.

4. μ_s( Cbν,υ2 ∪ P_r–3) ≤ 1; (ν, υ) ≠ (2, 3)

   where

   r=4+2υ(ν2−1)+2⌊υ−14⌋+4⌊υ−24⌋ ;ν≡0(mod2)6+2υ(ν−32)+6⌊υ−14⌋+4⌊υ−24⌋ ;ν≡1(mod2)

Figure 5

SEMT forest Cb4,52 ∪ P₁₅

Proof

1. Consider the graph ℘ ≅ Cbν,υ2 ∪ P_ϱ

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

   p=νυ+ϱ+1q=νυ+ϱ−1

   where

   ϱ=r   ;fora(i)r−1 ;fora(ii)

   Before formulating the labeling Ψ : V(℘) → {1, 2, …, p}, keep in view the labeling Υ defined in Theorem 2.

   For υ−12 odd;

   Take the previously mentioned labeling Υ (for υ−12 odd) with A=⌊νυ2⌋+⌈ϱ−12⌉. We define the labeling Ψ as follows:

   Ψ(xt)=⌊νυ2⌋+k   ;t=2k,1≤k≤⌈ϱ−12⌉νυ+⌊νυ2⌋+k−υ+3(υ−34)+2 ;t=2k−1,1≤k≤ϱ2,fora(i)νυ+⌊νυ2⌋+k−υ+3(υ−34)+1 ;t=2k−1,1≤k≤⌈ϱ2⌉,fora(ii)

   Ψ(x_ı,ȷ) = Υ(x_ı,ȷ) ; 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν

   Ψ(xυ+12,0)=νυ+⌊νυ2⌋+ϱ2−υ+3(υ−34)+3 ;fora(i)νυ+⌊νυ2⌋+⌈ϱ2⌉−υ+3(υ−34)+2 ;fora(ii)

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ + 1, ħ + 2, …, ħ + q}, where ℏ=⌊νυ2⌋+⌈ϱ−12⌉+1. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + min(S), where min(S) = ħ + 1.

   For υ−12 even;

   Take the labeling Υ which we have defined in Theorem 2 (for υ−12 even) with A′=⌈νυ2⌉+⌈ϱ−12⌉. We define the labeling Ψ as follows:

   Ψ(xt)=⌈νυ2⌉+k   ;t=2k,1≤k≤⌈ϱ−12⌉νυ+⌈νυ2⌉+k−υ+3(υ−54)+3 ; t=2k−1,1≤k≤ϱ2,fora(i)νυ+⌈νυ2⌉+k−υ+3(υ−54)+2 ; t=2k−1,1≤k≤⌈ϱ2⌉,fora(ii)

   Ψ(x_ı,ȷ) = Υ(x_ı,ȷ); 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν

   Ψ(xυ+12,0)=νυ+⌈νυ2⌉+ϱ2−υ+3(υ−54)+4;fora(i)νυ+⌈νυ2⌉+⌈ϱ2⌉−υ+3(υ−54)+3 ;fora(ii)

   The edge-sums generated by the above labeling ”Ψ” are the set of consecutive positive integers S = {ħ′ + 1, ħ′ + 2, …, ħ′ + q}, where ℏ′=⌈νυ2⌉+⌈ϱ−12⌉+1. Thus by Lemma 1, ”Ψ” can be extended to a SEMT labeling of ℘ and we obtain the magic constant a = p + q + min(S), where min(S) = ħ′ + 1.

2. Let ℧≅Cbν,υ2∪Pϱ∪K1 , where

   V(K1)={z}

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

   p′=νυ+ϱ+2

   q′=νυ+ϱ−1

   where

   ϱ=r−2  ;forb(i)r−3   ;forb(ii)

   Before formulating the labeling Ψ′ : V(℧) → {1, 2, …, p′}, keep in view the labeling Υ defined in Theorem 2.

   For υ−12 odd;

   Υ(x_ı,ȷ) = Ψ(x_ı,ȷ) = Ψ′(x_ı,ȷ); 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν, for b(i) and b(ii) both

   with A=⌊νυ2⌋+⌈ϱ−12⌉

    Ψ′(xt)=Ψ(xt),t≡0(mod2)Ψ′(xt)=νυ+⌊νυ2⌋+k−υ+3(υ−34)+1  ;t=2k−1,1≤k≤⌊ϱ+12⌋,forb(i)νυ+⌊νυ2⌋+k−υ+3(υ−34)  ;t=2k−1,1≤k≤ϱ+12,forb(ii)

   Let B=νυ+⌊νυ2⌋+⌊ϱ+12⌋−υ+3(υ−34)+1

   and C=νυ+⌊νυ2⌋+ϱ+12−υ+3(υ−34) , then

     Ψ′(z)=B+1  ;forb(i)C+1  ;forb(ii)Ψ′(xυ+12,0)=B+2  ;forb(i)C+2  ;forb(ii)

   The edge-sums generated by the above labeling “Ψ′” are the set of consecutive positive integers S = {ћ + 1, ћ + 2, …, ћ + q′}, where ℏ=⌊νυ2⌋+⌈ϱ−12⌉+1 . Thus by Lemma 1, “Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + min(S), where min(S) = ћ + 1.

   For υ−12 even;

   Υ(x_ı,ȷ) = Ψ(x_ı,ȷ) = Ψ′(x_ı,ȷ); 1 ≤ ı ≤ υ, 1 ≤ ȷ ≤ ν, for b(i) and b(ii) both

   with A′=⌈νυ2⌉+⌈ϱ−12⌉

   Ψ′(xt)=Ψ(xt),t≡0(mod 2)

   Ψ′(xt)=νυ+⌈νυ2⌉+k−υ+3(υ−54)+2 ;t=2k−1,1≤k≤⌊ϱ+12⌋,forb(i)νυ+⌈νυ2⌉+k−υ+3(υ−54)+1 ;t=2k−1,1≤k≤ϱ+12,forb(ii)

   Let B=νυ+⌈νυ2⌉+⌊ϱ+12⌋−υ+3(υ−54)+2

   and C=νυ+⌈νυ2⌉+ϱ+12−υ+3(υ−54)+1 , then

   Ψ′(z)=B+1 ;forb(i)C+1 ;forb(ii)

   Ψ′(xυ+12,0)=B+2 ;forb(i)C+2 ;forb(ii)

   The edge-sums generated by the above labeling “Ψ′” are the set of consecutive positive integers S = {ћ′ + 1, ћ′ + 2, …, ћ′ + q′}, where ℏ′=⌈νυ2⌉+⌈ϱ−12⌉+1 . Thus by Lemma 1, “Ψ′” can be extended to a SEMT labeling of ℧ and we obtain the magic constant a = p′ + q′ + min(S), where min(S) = ћ′ + 1. □