Generalized Symmetric Graphs

A graPh is called End-regular if its endomorphism semigrouP EndX isregular in the sense that for each f in EndX there.exists g in EndXwith fgf = f. In this paPer, End-regular circulant graPhs of degrees lessthan four are characterized. A graPh is called a weakly veftex-transitivegraPh(or vertex-symmetric graPh)if the endomorphism monoid actstransitively on the vertex set. The weakly edge-transitive graPh (weaklyt-arcs-transitive graPh and weakly distance-transitive graPh) is similarlydefined. In this paPer, some propefties of graPhs with weak transitionsare obtained. Moreovef, we use them to decide whether some graPhs areweakly transitive graPhs or not.