Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly det...Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly determine all commutative rings R(up to isomorphism)such that CL(R)has genus at most two.It is shown that for any Artinian ring R,CL(R)is a projective graph if and only if R is isomorphic to F4.Furthermore,we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.展开更多
The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less than 2 β(G?1/2 β(G?1 β(G) and not larger than β(G). Furt...The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less than 2 β(G?1/2 β(G?1 β(G) and not larger than β(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.展开更多
基金supported by Dr.D.S.Kothari Postdoctoral Fellowship Scheme(No.F.4-2/2006(BSR)/MA/17-18/0045)of University Grants Commission,India,awarded to the first authorthe second author was supported by UGC SAP DRS-1(No.F.510/7/DRS-1/2016(SAP-I),University Grants Commission,Government of IndiaThe research of S.Pirzada was supported by SERB-DST research project number CRG/2020/000109.
文摘Let R be a ring(not necessarily commutative)with identity 1 and let CL(R)be its clean graph.In this paper,we investigate the genus number of the compact Riemann surface in which CL(R)can be embedded and explicitly determine all commutative rings R(up to isomorphism)such that CL(R)has genus at most two.It is shown that for any Artinian ring R,CL(R)is a projective graph if and only if R is isomorphic to F4.Furthermore,we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.
基金This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 60373030,10751013)
文摘The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less than 2 β(G?1/2 β(G?1 β(G) and not larger than β(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.
