The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
Hartsfield and Ringel conjectured that every connected graph other than K2 is antimagic.Since then,many classes of graphs have been proved to be antimagic.But few is known about the antimagicness of lexicographic prod...Hartsfield and Ringel conjectured that every connected graph other than K2 is antimagic.Since then,many classes of graphs have been proved to be antimagic.But few is known about the antimagicness of lexicographic product graphs.In this paper,via the construction of a directed Eulerian circuit,the Siamese method,and some modification on graph labeling,the antimagicness of lexicographic product graph G[Pn]is obtained.展开更多
文摘The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
基金supported by the National Natural Science Foundation of China (Nos. 11401430)
文摘Hartsfield and Ringel conjectured that every connected graph other than K2 is antimagic.Since then,many classes of graphs have been proved to be antimagic.But few is known about the antimagicness of lexicographic product graphs.In this paper,via the construction of a directed Eulerian circuit,the Siamese method,and some modification on graph labeling,the antimagicness of lexicographic product graph G[Pn]is obtained.
