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This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view.

Zagreb indices --- domination --- Devaney chaos --- rotationally-symmetric convex polytopes --- topologically mixing --- geometric arithmetic index --- connectivity --- Metric dimension --- atom-bond connectivity ABC index --- distinguishing number --- gear graph --- titanium difluoride --- algorithm --- generalized gear graph --- general Randi? index --- cuprite --- atom bond connectivity index --- binary locating-domination number --- basis --- hypercyclicity --- functigraph --- complete graph --- alpha-boron nanotube --- general randi? index --- geodesics --- inverse degree index --- Hex-Derived Cage networks --- metric dimension --- dominating set --- bipartite graphs --- products of graphs --- geometric-arithmetic GA index --- direct product of graphs --- gromov hyperbolicity --- secure resolving set and secure resolving domination --- topological indices --- graph operators --- resolving set --- harmonic index --- polycyclic aromatic hydrocarbons --- ILP models --- disjointness --- topological transitivity --- Gromov hyperbolicity --- metric basis --- harmonic polynomial

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Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz's Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites.Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called "the plaid model," has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics.The book includes an extensive computer program that allows readers to explore materials interactively and each theorem is accompanied by a computer demo.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- 1-dimensional wordlines. --- Bad Tile Lemma. --- Box Theorem. --- Copy Lemma. --- Copy Theorem. --- Curve Turning Theorem. --- Graph Master Picture Theorem. --- Graph Master Theorem. --- Graph Reconstruction Lemma. --- Grid Geometry Lemma. --- Grid Supply Lemma. --- Horizontal Lemma. --- Intertwining Lemma. --- Master Picture Theorem. --- Matching Criterion. --- Orbit Equivalence Theorem. --- PET. --- Plaid Master Picture Theorem. --- Projection Theorem. --- Quasi-Isomorphism Theorem. --- Rectangle Lemma. --- Renormalization Theorem. --- Segment Lemma. --- Truchet Comparison Theorem. --- Truchet tile system. --- Vertical Lemma. --- anchor point. --- arithmetic alignment. --- arithmetic graph. --- auxiliary lemmas. --- capacities. --- capacity sequence. --- capacity. --- checkerboard partition. --- classifying map. --- compactification. --- congruence. --- continued fractions. --- convex polygon. --- convex polytopes. --- corner percolation. --- east edges. --- elementary number theory. --- embedded loops. --- equidistribution. --- even rational parameter. --- fundamental surface. --- geometric alignment. --- geometry. --- graph grid. --- grid lines. --- horizontal case. --- integer square. --- intersection points. --- kites. --- kits. --- lemma. --- lemmas. --- light point. --- light points. --- linear algebra. --- low capacity lines. --- map. --- maps. --- mass sequence. --- masses. --- number theory. --- orbit. --- oriented lines. --- outer billiards map. --- outer billiards. --- parallel lines. --- parallelotope. --- particle lines. --- pixelated spacetime diagrams. --- pixelated spacetime. --- pixilation. --- plaid PET map. --- plaid model. --- plaid polygon. --- plaid polygons. --- plane. --- planetary motion. --- planetary orbits. --- polygon. --- polytope exchange transformations. --- polytopes. --- prism structure. --- prism. --- proof. --- quarter turn compositions. --- remote adjacency. --- renormalization. --- scale information. --- sequences. --- slanting lines. --- spaces. --- spacetime diagrams. --- spacetime plaid surfaces. --- special billiards orbits. --- square tiling. --- stacking blocks. --- structural result. --- symmetries. --- symmetry. --- technical lemma. --- theorems. --- vertical case. --- vertical particles.