• Journal of Applied and Pure Mathematics
• /
• v.5 no.3_4
• /
• pp.237-253
• /
• 2023

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}=\{\array{{\frac{p}{2}} & \;\;p\text{ is even} \\ {\frac{p-1}{2}} & \;\;p\text{ is odd,}$$ and M = {±1, ±2, … ± ρ} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling ${\frac{{\lambda}(u)+{\lambda}(v)}{2}}$ if λ(u) + λ(v) is even and ${\frac{{\lambda}(u)+{\lambda}(v)+1}{2}}$ if λ(u) + λ(v) is odd such that ${\mid}{\bar{{\mathbb{S}}}}_{\lambda}{_1}-{\bar{{\mathbb{S}}}}_{{\lambda}^c_1}{\mid}{\leq}1$ where ${\bar{{\mathbb{S}}}}_{\lambda}{_1}$ and ${\bar{{\mathbb{S}}}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of few graphs including the closed helm graph, web graph, jewel graph, sunflower graph, flower graph, tadpole graph, dumbbell graph, umbrella graph, butterfly graph, jelly fish, triangular book graph, quadrilateral book graph.

• Journal of Applied and Pure Mathematics
• /
• v.6 no.1_2
• /
• pp.55-69
• /
• 2024

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}=\{\array{{\frac{p}{2}} && p\;\text{is even} \\ {\frac{p-1}{2}} && p\;\text{is odd,}}$$ and M = {±1, ±2, … ± 𝜌} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling $\frac{{\lambda}(u)+{\lambda}(v)}{2}$ if λ(u) + λ(v) is even and $\frac{{\lambda}(u)+{\lambda}(v)+1}{2}$ if λ(u) + λ(v) is odd such that ${\mid}\bar{\mathbb{s}}_{{\lambda}_1}-\bar{\mathbb{s}}_{{\lambda}^c_1}{\mid}\,{\leq}\,1$ where $\bar{\mathbb{s}}_{{\lambda}_1}$ and $\bar{\mathbb{s}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G with a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of some union of graphs.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.1
• /
• pp.69-77
• /
• 2006

The generalized Hyers-Ulam stability problems of the mixed type functional equation $$f\({\sum_{i=1}^{4}xi\)+\sum_{1{\leq}i<j{\leq}4}f(x_i+x_j)=\sum_{i=1}^{4}f(x_i)+\sum_{1{\leq}i<j<k{\leq}4}f(x_i+X_j+x_k)$$ is treated under the approximately even(or odd) condition and the behavior of the quadratic mappings and the additive mappings is investigated.

• Journal of applied mathematics & informatics
• /
• v.26 no.3_4
• /
• pp.583-590
• /
• 2008

In this article, we focuss on. sequences of polynomials with {$\pm1$} coefficients constructed by recursive argument that is known as Rudin-Shapiro polynomials. The asymptotic behavior of these polynomials defines as the ratio of their 2q-norm with 2-norm to be dominated by some number depending on q or "the best" by an absolute constant. In this work we first show the conjecture holds for some finite numbers of m and then introduce a technique that give the result for any positive odd integer m whenever it holds for all pervious even numbers.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.2
• /
• pp.189-196
• /
• 2006

The generalized Hyers-Ulam stability problems of the cubic functional equation f(x + y + z) + f(x + y - z) + 2f(x - y) + 4f(y) = f(x - y + z) + f(x - y - z) +2f(x + y) + 2f(y + z) + 2f(y - z) shall be treated under the approximately odd condition and the behavior of the cubic mappings and the additive mappings shall be investigated. The generalized Hyers-Ulam stability problem for functional equations had been posed by Th.M. Rassias and J. Tabor [7] in 1992.

• Applied Chemistry for Engineering
• /
• v.21 no.2
• /
• pp.230-237
• /
• 2010

A homologous series of linear liquid crystal dimers, ${\alpha},{\omega}$-bis(4-nitroazobenzene-4'-carbonyloxy)alkanes (NATWESn, n = 2~8, 10, the number of methylene units in the spacer) have been synthesized, and the thermal behavior of the series has been investigated. All the dimers formed enantiotropic nematic phases. The nematic-isotropic transition temperatures of the dimers and their entropy variation at the phase transition showed a large odd-even effect as a function of n. This behavior was rationalized in terms of the change in the average shape of the spacer on varing the parity of the spacer. The thermal stability and degree of order in the nematic phase and the magnitude of the odd-even effect of NATWESn were very similar to those of the corresponding ether compounds, while they were significantly different from those of the monomesogenic compounds, 4-{4'-(nitrophenylazo)phenoxy}alkanoyl chlorides and the side-chain liquid-crystalline polymers, the poly[1-{4-(4'-nitrophenylazo) phenoxycarbonylalkanoyloxy}ethylene]s. The results were discussed in terms of the 'irtual trimer model'by Imrie.

• Journal of applied mathematics & informatics
• /
• v.42 no.1
• /
• pp.1-14
• /
• 2024

Let G = (V, E) be a (p, q) graph. Define $$\begin{cases}\frac{p}{2},\:if\:p\:is\:even\\\frac{p-1}{2},\:if\:p\:is\:odd\end{cases}$$ and L = {±1, ±2, ±3, ···, ±ρ} called the set of labels. Consider a mapping f : V → L by assigning different labels in L to the different elements of V when p is even and different labels in L to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that |Δ_f1 - Δ_f^c1| ≤ 1, where Δ_f1 and Δ_f^c1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of subdivision of some graphs.

• Applied Chemistry for Engineering
• /
• v.22 no.4
• /
• pp.358-366
• /
• 2011

A homologous series of linear liquid crystal dimers, the ${\alpha},{\omega}$-bis(4-cyano-azobenzene-4'-oxy)alkanes (CATWETn, where n, the number of methylene units in the spacer, is 2~10) were synthesized, and their thermotropic liquid crystalline phase behavior were investigated. The CATWETn with n of 3 and 6 exhibited monotropic nematic phases, whereas other derivatives showed enantiotropic nematic phases. The nematic-isotropic transition temperatures of the dimers and their entropy variation at the phase transition showed a large odd-even effect as a function of n. This phase transition behavior was rationalized in terms of the change in the average shape of the spacer on varying the parity of the spacer. The thermal stability and degree of order in the nematic phase and the magnitude of the odd-even effect of CATWETn were similar to those for the methoxy-, nitro-, and pentyl-substituted dimers, while they were significantly different from those for the monomesogenic compounds, 1-{4-(4'-cyanophenylazo)phenoxy}alkylbromides and the side-chain liquid-crystalline polymers, the poly[1-{4-(4'-cyanophenylazo)phenoxyalkyloxy}ethylene]s. The results were discussed in terms of 'virtual trimer model' by Imrie.

• Bulletin of the Korean Chemical Society
• /
• v.28 no.12
• /
• pp.2241-2247
• /
• 2007

A new series of chiral bent-shaped liquid crystals with an asymmetric central core based on 1,6- dihydroxynaphthalene and chiral terminal chain prepared from (S)-(?)-2-methyl-1-butanol, 1,6-naphthalene bis[4-(4-alkoxyphenyliminomethyl)]benzoates [N(1,6)-n-O-PIMB(n-2)*-(n-4)O (n = 8-11)] were synthesized. Their mesomorphic properties and phase structures were investigated by means of electro-optical, polarization reversal current, and second harmonic generation measurements in order to confirm the relationship between the molecular structure and phase structure. All odd n (n = 9 and 11) compounds, N(1,6)-9-O-PIMB7*-5O and N(1,6)-11-O-PIMB9*-7O exhibit antiferroelectric phase, whereas even n (n = 8 and 10) compounds was flexible, N(1,6)-10-O-PIMB8*-6O exhibits the ferroelectric phase but N(1,6)-8-O-PIMB6*-4O exhibits the antiferroelectric phase. These results come from the decrease of the closed packing efficiency within a layer and the lack of uniform interlayer interaction between adjacent layers, which were caused by the asymmetrical naphthalene central core. Thus, we concluded that the structure of central core as well as the terminal chain plays an important role for the emergence of particular polar ordering in phase structures.

• Polymer(Korea)
• /
• v.32 no.5
• /
• pp.489-496
• /
• 2008

The thermotropic liquid crystalline behavior of a homologous series of poly[1-{4-(4' nitrophenylazo) phenoxycarbonylalkanoyloxy}ethylene]s (NAPEn, n = $2{\sim}8$,10, the number of methylene units in the spacer) have been investigated. All of the homologues formed monotropic nematic phases. The glass transition temperatures decreased with n. This is attributed to a plasticization of the backbone by the side chains. The isotropic-nematic phase transition temperatures decreased with increasing n up to 7 and showed the odd-even effect. However it became almost constant when n is more than 7. This behavior was rationalized in terms of the change in the average shape of the side chain on varing the parity of the spacer. This rationalization also accounts for the observed variation of the entropic gain for the clearing transition. The mesophase properties of NAPEn were entirely different from those reported for the polymers in which the azobenzene groups are attached to polyacrylate, polymathacrylate, and polystyrene backbones through polymethylene spacers. The results indicate that the mode of chemical linkage of the side group with the main chain plays an important role in the formation, stabilization, and type of mesophase.