چكيده به لاتين
Vertex-transitive, edge-transitive and arc-transitive are algebraic properties, and the distance between two vertices is the metric property of graphs. There are lots of classification of graphs based on these properties. The aim of this thesis, is classification of finite, connected cubic and pentavalent arc-transitive and also vertex-transitive distance balanced graphs. So, first, pentavalent symmetric graphs of orders 18p and 8pq have been classified, for two prime inte- gers p and q. Also, cubic symmetric graphs of order 18p^2 are classified. Next, by considering integers a ∈ {3,...,8} and b ∈ {1,...,4}, have been studied pentavalent strongly connected 1−transitive digraphs of order 2^a.p^b.q, with insolvable automorphism group G. Therefore, after characterizing all possible groups G, was presented a tentatively complete classification of these digraphs. Furthermore, by investigating the properties of nicely distance balanced graphs, as a generalization of distance balanced graphs, was determined the complete classification of vertex-transitive nicely distance balanced graphs X with consistent number γX= 4.
