Some Properties Of Transformation Graphs

Recently some scholars study transformation graphs and present a lot of results in succession—the sufficient and necessary condition of connected transformation graphs, for example. By now, results of transformation graphs are less. Thus there are a lot of properties of transformation graphs deserving to study. And then we study regular and spectral properties of transformation graphs. At the same time, we present an upper bound on spectral radii of total graphs of trees. The conclusions are as follows:1. The regularity of transformation graphs(Chap. 2):G⁺⁺⁺ and G^--- are regular if and only if G is regular; G^++- and G^--+ are regular if and only if G = G_n or K_2,n-2 or K₄ G^+-+ and G^-+- are regular if and only if G (?) C₅ or K₇ or K₂ or K_3,3 or G₀ (see Graph 1); G^-++ and G^+--- are regular if and only if G is (n-1)/2 -regular.2. Some upper bounds on the spectral radius of transformation graphs (In Chap. 3, n is the number of vertices in G, m is the number of edges in G):3. In Chap. 4 we show that total graphs of stars achieve uniquely maximal spectral radii in total graphs of all trees.