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Given graph  G ( V , E ) . We use the notion of total  k -labeling which is edge irregular. The notion  of total edge irregularity strength (tes) of graph  G  means the minimum integer  k  used in the edge irregular total k-labeling of  G . A cactus graph  G  is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle  C n  with same size  n  is named an  n -uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then  G  is called  n -uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs  C ( C n r ) of length  r  for some  n  ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs T r (4, n ) and T r (5, n ) of length r . Our results are as follows: tes( C ( C n r )) = ⌈( nr + 2)/3⌉ ; tes( T r (4, n )) = ⌈((5+ n ) r +2)/3⌉ ; tes( T r (5, n )) = ⌈((5+ n ) r +2)/3⌉.

Computing total edge irregularity strength of some n-uniform cactus chain graphs and related chain graphs