This monograph is for theoty and its extensions related to census of maps on general surfaces estabhshed on the basis of what has been done on the surface of genus zero.In spite of a number of improved results with maps on a surface of genus zero for surfaces of genus not zero. two new theoretical lines wia exploiting the relationship between super maps and embeddings of a graph on surfaces and via the joint tree technique are investigated for a variety of topics such as those in the determination of handle and crosscap polynomials of maps, genus distribution of embeddings, and others related. In particular, an appendix serves as the exhaustive counting super maps (rooted and nonrooted) including these polynomials with under graphs of small size for the reader\\\'s digests. Although the book is mainly for researchers in mathematics, theoretical physics, chemistry, biology and others related, the basic part in each chapter can also be chosen for graduates and college teachers as references.
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目录
Preface Chapter 1 Preliminaries §1.1 Maps §1.2 Polynomials on maps §1.3 Enufunctions §1.4 Polysum functions §1.5 The Lagrangian inversion §1.6 The shadow functional §1.7 Asymptotic estimation §1.8 Notes Chapter 2 Outerplanar Maps §2.1 Plane trees §2.2 Wintersweets §2.3 Unicyclic maps §2.4 General outerplanar maps §2.5 Notes Chapter 3 Triangulations §3.1 Outerplanar triangulations §3.2 Planar triangulations §3.3 Triangulations on the disc §3.4 Triangulations on the projective plane §3.5 Triangulations on the torus §3.6 Notes Chapter 4 Quadrangulations §4.1 Outerplanar quadrangulations §4.2 Outerplanar quadrangulations on the disc §4.3 Hamiltonian quadrangulations on the sphere §4.4 Inner endless planar quadrangulations §4.5 Quadrangulations on the projective plane §4.6 Quadrangulations on the Klein bottle §4.7 Notes Chapter 5 Eulerian Maps §5.1 Planar Eulerian maps §5.2 Tutte formula §5.3 Eulerian planar triangulations §5.4 Regular Eulerian planar maps §5.5 Eulerian maps on surfaces §5.6 Notes Chapter 6 Nonseparable Maps §6.1 Outerplanar nonseparable maps §6.2 Eulerian nonseparable maps §6.3 Planar nonseparable maps §6.4 Nonseparable maps on surfaces §6.5 Bridgeless maps on surfaces §6.6 Notes Chapter 7 Simple Maps §7.1 Loopless maps §7.2 General simple maps §7.3 Simple bipartite maps §7.4 Loopless maps on surfaces §7.5 Notes Chapter 8 General Maps §8.1 General planar maps §8.2 Planar c-nets §8.3 Convex polyhedra §8.4 Quadrangulations via c-nets §8.5 General maps on surfaces §8.6 Notes Chapter 9 Chrosum Equations §9.1 Tree equations §9.2 Outerplanar equations §9.3 General equations §9.4 Triangulation equations §9.5 Well de¯nedness §9.6 Chrosums on surfaces §9.7 Notes Chapter 10 Polysum Equations §10.1 Polysums for bitrees §10.2 Outerplanar polysums §10.3 General polysums §10.4 Nonseparable polysums §10.5 Polysums on surfaces §10.6 Notes Chapter 11 Maps via Embeddings §11.1 Automorphism group of a graph §11.2 Embeddings of a graph §11.3 Super maps of a graph §11.4 Maps from embeddings §11.5 Notes Chapter 12 Locally Oriented Maps §12.1 Planar Hamiltonian maps §12.2 Biboundary inner rooted maps §12.3 Boundary maps §12.4 Cubic boundary maps §12.5 Notes Chapter 13 Genus Polynomials of Graphs §13.1 Joint tree model §13.2 Layer divisions §13.3 Graphs from smaller §13.4 Pan-bouquets §13.5 Notes Chapter 14 From Rooted to Unrooted §14.1 Symmetric relations §14.2 An application §14.3 Symmetric principles §14.4 General examples §14.5 From under graphs §14.6 Notes Chapter 15 From Planar to Nonplanar §15.1 Trees with boundary §15.2 Cutting along vertices §15.3 Cutting along faces §15.4 Maps with a plane base §15.5 Vertex partition §15.6 Notes Chapter 16 Chromatic Solutions §16.1 General solution §16.2 Cubic triangles §16.3 Invariants §16.4 Four color solutions §16.5 Notes Chapter 17 Stochastic Behaviors §17.1 Asymptotics for outerplanar maps §17.2 The average on tree-rooted maps §17.3 Hamiltonian circuits per map §17.4 The asymmetry on maps §17.5 Asymptotics via equations §17.6 Notes Appendix Atlas of Super Maps for Small Graphs Ax.1 Bouquets Bm, 4≥m≥1 Ax.2 Link bundles Lm, 6≥m≥3 Ax.3 Complete bipartite graphs Km；n；4≥m；n≥3 Ax.4 Wheels Wn, 5≥n≥4 Ax.5 Triconnected cubic graphs of size in [6；15] Bibliography Subject Index Author Index
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