A graph is called a semi-regular graph if its automorphism group action onits ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficientcondition for an automorphism of the graph Γ t...A graph is called a semi-regular graph if its automorphism group action onits ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficientcondition for an automorphism of the graph Γ to be an automorphism of a map with the underlyinggraph Γ is obtained. Using this result, all orientation-preserving automorphisms of maps onsurfaces (orientable and non-orientable) or just orientable surfaces with a given underlyingsemi-regular graph Γ are determined. Formulas for the numbers of non-equivalent embeddings of thiskind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, thenon-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable andgeneral surfaces are enumerated.展开更多
Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means ...Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas. 展开更多
We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalizatio...We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.展开更多
We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in...We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.展开更多
It was shown by Formanek and Sibley that the group determined characterizes a finite groupG up to isomorphism. Hoehnke and Johnson (independelltly the suthors--using an argumentof Manslield) showed the corresponding r...It was shown by Formanek and Sibley that the group determined characterizes a finite groupG up to isomorphism. Hoehnke and Johnson (independelltly the suthors--using an argumentof Manslield) showed the corresponding result for k-characters, k = 1, 2, 3. The notion of kcharacters dates back to nobenius. They are determined by the group doterminaDt and maybe derived from the character table CT(G) provided one knows additionally the functionswhere C(C) = {Cg, g E G} denotes the set of conjugacy classes of G.The object of the paper is to present criteria for finite groups (more precisely for solublegroups G and H which are both semi-direct products of a similar type) when1. G and H have isomorphic spectral tables (i.e., they form a Brauer pair),2. G and H have isomorphic table of marks (in particular the Burnside rings are isomorphic),3. G and H have the same 2-characters.Using this the authors construct two non-iS.Omorphic soluble groups for which all these threerepresent at iont heor et ical invar taut s coincide.展开更多
基金The first and the second authors are partially supported by NNSFC under Grant No.60373030The third author is partially supported by NNSFC under Grant No.10431020
文摘A graph is called a semi-regular graph if its automorphism group action onits ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficientcondition for an automorphism of the graph Γ to be an automorphism of a map with the underlyinggraph Γ is obtained. Using this result, all orientation-preserving automorphisms of maps onsurfaces (orientable and non-orientable) or just orientable surfaces with a given underlyingsemi-regular graph Γ are determined. Formulas for the numbers of non-equivalent embeddings of thiskind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, thenon-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable andgeneral surfaces are enumerated.
文摘Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.
文摘We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.
文摘We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.
文摘It was shown by Formanek and Sibley that the group determined characterizes a finite groupG up to isomorphism. Hoehnke and Johnson (independelltly the suthors--using an argumentof Manslield) showed the corresponding result for k-characters, k = 1, 2, 3. The notion of kcharacters dates back to nobenius. They are determined by the group doterminaDt and maybe derived from the character table CT(G) provided one knows additionally the functionswhere C(C) = {Cg, g E G} denotes the set of conjugacy classes of G.The object of the paper is to present criteria for finite groups (more precisely for solublegroups G and H which are both semi-direct products of a similar type) when1. G and H have isomorphic spectral tables (i.e., they form a Brauer pair),2. G and H have isomorphic table of marks (in particular the Burnside rings are isomorphic),3. G and H have the same 2-characters.Using this the authors construct two non-iS.Omorphic soluble groups for which all these threerepresent at iont heor et ical invar taut s coincide.
