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On oriented $2$-factorable graphs
Linfan Mao,Feng Tian 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1-2
The enumeration problem in graph theory, including the enumeration of labelled graphs, graphs, rooted maps on surface, embeddings of a graph and unrooted maps on surface, has been intensively investigated by mathematicians in the past century and appear many excellent survey books on dierent branches, such as Polya. G and R. C. Read’s [9], Tutte’s [11], Harary. F and Palmer. E. M’s [2], Liu’s [14], · · · . The essence of this problem is to count the non-isomorphic objects under dierent actions of group, for example, the enumeration of graphs of order n means to count the non-isomorphic graphs of order n under the action of graph isomorphisms. As we known, dierent objects of enumeration use dierent approaches. Generally, three main techniques are used in solving this problem.
GROUP ACTION FOR ENUMERATING MAPS ON SURFACES
Mao, Linfan,Liu, Yanpei 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.13 No.1
A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. For any connected graph $\Gamma$, by introducing the concertion of semi-arc automorphism group Aut$\_$$\frac{1}{2}$/$\Gamma$ and classifying all embedding of $\Gamma$ undo. the action of this group, the numbers r$\^$O/ ($\Gamma$) and r$\^$N/($\Gamma$) of rooted maps on orientable and non-orientable surfaces with underlying graph $\Gamma$ are found. Many closed formulas without sum ∑ for the number of rooted maps on surfaces (orientable or non-orientable) with given underlying graphs, such as, complete graph K$\_$n/, complete bipartite graph K$\_$m, n/ bouquets B$\_$n/, dipole Dp$\_$n/ and generalized dipole (equation omitted) are refound in this paper.
ON ORIENTED 2-FACTORABLE GRAPHS
MAO, LINFAN,TIAN, FENG 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1
Oriented 2-factorable graphs are reduced to bouquets by permutation voltage assignment in this paper. Introducing the concept of k-class index of a permutation group, various oriented 2-factorable graphs are enumerated in this paper.
Invariance problem in structural non-probabilistic reliability index
Xinzhou Qiao,Linfan Song,Peng Liu,Xiurong Fang 대한기계학회 2021 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.35 No.11
The non-probabilistic reliability index has been extensively used to evaluate the safety degree of structures with limited experiment data. By dealing the uncertain parameters of structures with the ellipsoidal model, this paper investigates the invariance problem in the nonprobabilistic reliability index. A prerequisite of the existence of the invariance problem is first given. An investigation of whether the two non-probabilistic first order reliability methods, namely the mean-value and design-point methods, encounter the same problem is then presented. A comparison of the precision of these two methods is further conducted through three numerical examples, and based on which some significant phenomena are summarized.