核的圈秩为2的图的边色数

设Ｇ是简单图．Ｇ的最大度点的导出子图称为Ｇ的核．本文讨论核的圈秩为２的简单图的边着色分类问题．

合成图的全着色

对于任意简单图G,Δ(G)和t(G)分别表示G的最大度和全色数.本文证明了如果G的全色数满足t(G)≤Δ(G)+2,则合成图G[(?)_m]和K_n[G]的全色数满足t(G[(?)_m])≤Δ(G[(?)_m])+2,t(K_n[G])≤Δ(K_n[G])+2。

Chromatically Equivalent Graphs with P(G, λ) = (λ- λ_1)(λ-λ_2)λ￣(n+1)

The author finds out all of the graphs with the same chromatic polynomial and proves some relevant theorems and corollaries.

Leptin and peroxisome proliferator-activated receptor γ expression in colorectal adenoma

AIM： To investigate the expressions of leptin and per- oxisome proliferator-activated receptor y （PPARG） in relation to body mass index （BMI）. METHODS： We evaluated leptin and PPARG expres- sion in 30 adenomas over 1 cm in size by immunohisto- chemical staining. In addition, clinicopathologic features including BMI were assessed. RESULTS： PPARG and leptin expression showed a strong positive correlation （P = 0.035）. The average BMI of the leptin-positive group was higher than that of the leptin-negative group （25.4 ＋ 3.4 kg/m2 vs 22.6 ＋ 2.4 kg/m2, P = 0.018）, and leptin expression was sig- nificantly correlated with high BMI （P = 0.024）. Leptinexpression was more frequently observed in intermedi- ate/high grade dysplasia than in low grade dysplasia （P = 0.030）. However, PPARG expression was not cor- related with BMI and grade of dysplasia. CONCLUSION： BMI has influenced on the leptin ex- pression of colorectal adenoma. The exact mechanism underlies the strong correlation between leptin and PPARG expression needs further study.