We obtain a new class of polynomial identities on the ring of n × n matrices over any commutative ring with 1 by using the Swan’s graph theoretic method [1] in the proof of Amitsur-Levitzki theorem. Let be an Eu...We obtain a new class of polynomial identities on the ring of n × n matrices over any commutative ring with 1 by using the Swan’s graph theoretic method [1] in the proof of Amitsur-Levitzki theorem. Let be an Eulerian graph with k vertices and d edges. Further let be an integer and assume that . We prore that is an PI on Mn(C). Standard and Chang [2] -Giambruno-Sehgal [3] polynomial identities are the spectial examples of our conclusions.展开更多
Hajos' conjecture asserts that a simple eulerian graph on n vertices can be decomposed into at most n-1/2 circuits. In this paper, we propose a new conjecture which is equivalent to Hajos' conjecture, and show...Hajos' conjecture asserts that a simple eulerian graph on n vertices can be decomposed into at most n-1/2 circuits. In this paper, we propose a new conjecture which is equivalent to Hajos' conjecture, and show that to prove Hajos' conjecture, it is sufficient to prove this new conjecture for 3-connected graphs. Furthermore, a special 3-cut is considered also.展开更多
A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）...A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）, then D is supereulerian. In this paper, we prove that if D is an extended cycle, an extended hamiltonian digraph, an arc-locally semicomplete digraph, an extended arc-locally semicomplete digraph, an extension of two kinds of eulerian digraph, a hypo-semicomplete digraph or an extended hypo-semicomplete digraph satisfying λ（D） ≥α（D）, then D is supereulerian.展开更多
The present research considers the problem of covering a graph with minimal number of trails satisfying the pre-defined local restrictions. The research is devoted to the problem of graph covering by minimal number of...The present research considers the problem of covering a graph with minimal number of trails satisfying the pre-defined local restrictions. The research is devoted to the problem of graph covering by minimal number of trails satisfying some local restrictions. Algotithm of allowed Eulerian cycle construction is considered. The authors showed that it is possible to recognize the system of transitions and solve the problem of constructing the allowable path by linear time. It’s also possible to find allowable Eulerian cycle for Eulerian graph or to proclaim that such a cycle does not exist by the time O(|V(G)|.|E(G)|). All presented algorithms have the software realization.展开更多
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamilt...The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.展开更多
文摘We obtain a new class of polynomial identities on the ring of n × n matrices over any commutative ring with 1 by using the Swan’s graph theoretic method [1] in the proof of Amitsur-Levitzki theorem. Let be an Eulerian graph with k vertices and d edges. Further let be an integer and assume that . We prore that is an PI on Mn(C). Standard and Chang [2] -Giambruno-Sehgal [3] polynomial identities are the spectial examples of our conclusions.
基金This research is partially supported by the National Postdoctoral Fund of China and Natural Science Foundation of China(No. 10001035) This work was finished while the author was working for Academy of Mathematics and Systems Science, Chinese of Academy
文摘Hajos' conjecture asserts that a simple eulerian graph on n vertices can be decomposed into at most n-1/2 circuits. In this paper, we propose a new conjecture which is equivalent to Hajos' conjecture, and show that to prove Hajos' conjecture, it is sufficient to prove this new conjecture for 3-connected graphs. Furthermore, a special 3-cut is considered also.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1176107161363020)+1 种基金Science and Technology Innovation Project of Xinjiang Normal University(Grant No.XSY201602013)the"13th Five-Year"Plan for Key Discipline Mathematics of Xinjiang Normal University(Grant No.17SDKD1107)
文摘A digraph D is supereulerian if D has a spanning eulerian subdigraph. Bang- Jensen and Thomasse conjectured that if the arc-strong connectivity ）,（D） of α digraph D is not less than the independence number α（D）, then D is supereulerian. In this paper, we prove that if D is an extended cycle, an extended hamiltonian digraph, an arc-locally semicomplete digraph, an extended arc-locally semicomplete digraph, an extension of two kinds of eulerian digraph, a hypo-semicomplete digraph or an extended hypo-semicomplete digraph satisfying λ（D） ≥α（D）, then D is supereulerian.
文摘The present research considers the problem of covering a graph with minimal number of trails satisfying the pre-defined local restrictions. The research is devoted to the problem of graph covering by minimal number of trails satisfying some local restrictions. Algotithm of allowed Eulerian cycle construction is considered. The authors showed that it is possible to recognize the system of transitions and solve the problem of constructing the allowable path by linear time. It’s also possible to find allowable Eulerian cycle for Eulerian graph or to proclaim that such a cycle does not exist by the time O(|V(G)|.|E(G)|). All presented algorithms have the software realization.
文摘The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
