In this note, we obtain a new method of proving a Cayley graph can whether or not be decomposed into Hamiltonian circuits and use this method, we prove that if a group G has some special properties, then Cayley graph ...In this note, we obtain a new method of proving a Cayley graph can whether or not be decomposed into Hamiltonian circuits and use this method, we prove that if a group G has some special properties, then Cayley graph (G,M) can be decomposed into two Hamiltonian circuits. This result answers a partial case of Alspach's conjecture concerning Hamiltonian decomposition of 2k-regular connected Cayley graphs.展开更多
Let S and K be two subrings of a finite ring R. Then the generalized non- commuting graph of subrings S, K of R, denoted by [＇S,K, is a simple graph whose vertex set is （S U K）／（CK（S） U Cs（K））, and where two...Let S and K be two subrings of a finite ring R. Then the generalized non- commuting graph of subrings S, K of R, denoted by [＇S,K, is a simple graph whose vertex set is （S U K）／（CK（S） U Cs（K））, and where two distinct vertices a, b are adjacent if and only if a E S or b E S and ab ≠ ba. We determine the diameter, girth and some dominating sets for FS, K. Some connections between Fs,K and Pr（S, K） are also obtained. Further, Z-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two Y_~isoclinic pairs are isomorphic under some conditions.展开更多
A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D.A total outer-independent dominating set of a graph G is a set D of vertices of G such that ...A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D.A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D,and the set V(G)\D is independent.The 2-domination(total outer-independent domination,respectively)number of a graph G is the minimum cardinality of a 2-dominating(total outer-independent dominating,respectively)set of G.We investigate the ratio between2-domination and total outer-independent domination numbers of trees.展开更多
Let ：T2k＋1 be the set of trees on 2k＋ 1 vertices with nearly perfect matchings, and let S2k＋2 be the set of trees on 2k ＋ 2 vertices with perfect matchings. The largest Laplacian spectral radii of trees in ：T2k...Let ：T2k＋1 be the set of trees on 2k＋ 1 vertices with nearly perfect matchings, and let S2k＋2 be the set of trees on 2k ＋ 2 vertices with perfect matchings. The largest Laplacian spectral radii of trees in ：T2k＋l and S2k＋2 and the corresponding trees were given by Guo （2003）. In this paper, the authors determine the second to the sixth largest Laplacian spectral radii among all trees in T2k＋1 and give the corresponding trees.展开更多
文摘In this note, we obtain a new method of proving a Cayley graph can whether or not be decomposed into Hamiltonian circuits and use this method, we prove that if a group G has some special properties, then Cayley graph (G,M) can be decomposed into two Hamiltonian circuits. This result answers a partial case of Alspach's conjecture concerning Hamiltonian decomposition of 2k-regular connected Cayley graphs.
文摘Let S and K be two subrings of a finite ring R. Then the generalized non- commuting graph of subrings S, K of R, denoted by [＇S,K, is a simple graph whose vertex set is （S U K）／（CK（S） U Cs（K））, and where two distinct vertices a, b are adjacent if and only if a E S or b E S and ab ≠ ba. We determine the diameter, girth and some dominating sets for FS, K. Some connections between Fs,K and Pr（S, K） are also obtained. Further, Z-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two Y_~isoclinic pairs are isomorphic under some conditions.
基金the Polish Ministry of Science and Higher Education grand IP/2012/038972
文摘A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D.A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D,and the set V(G)\D is independent.The 2-domination(total outer-independent domination,respectively)number of a graph G is the minimum cardinality of a 2-dominating(total outer-independent dominating,respectively)set of G.We investigate the ratio between2-domination and total outer-independent domination numbers of trees.
基金supported by the National Natural Science Foundation of China under Grant No. 10331020.
文摘Let ：T2k＋1 be the set of trees on 2k＋ 1 vertices with nearly perfect matchings, and let S2k＋2 be the set of trees on 2k ＋ 2 vertices with perfect matchings. The largest Laplacian spectral radii of trees in ：T2k＋l and S2k＋2 and the corresponding trees were given by Guo （2003）. In this paper, the authors determine the second to the sixth largest Laplacian spectral radii among all trees in T2k＋1 and give the corresponding trees.
