Total edge irregularity strength for special types of square snake graphs

Abstract

AbstractOne of the extremely useful branches in graph theory is the labeling of a graph. Graph labeling plays a vital role in many fields such as database management, astronomy, coding theory, X-ray crystallography, communication network addressing and radar. A labeling of a connected simple graph $$G\left( {V,E} \right)$$ G V , E is a map that assign each element in $$G$$ G with a positive integer number. An edge irregular total $$\lambda^{\!\!\!\!\!-}$$ λ - -labeling is a map $$\beta :V\left( G \right) \cup E\left( G \right) \to \left\{ 1,2,3, \ldots ,\lambda^{\!\!\!\!\!-} \right\}$$ β : V G ∪ E G → 1 , 2 , 3 , … , λ - such that $$W_\beta \left( h \right) \ne W_\beta \left( z \right)$$ W β h ≠ W β z where $$W_\beta \left( h \right)$$ W β h and $$W_\beta \left( z \right)$$ W β z are weights for any two distinct edges. In this case, $$G$$ G has total edge irregularity strength (TEIS) if $$\lambda^{\!\!\!\!\!-}$$ λ - is minimum. In this paper, a new family of graphs called square snake graphs is defined and denoted by $$C_{4,n}$$ C 4 , n . Moreover, we define some related graphs of square snake graphs named double square snake graph $$D\left( {C_{4,n} } \right)$$ D C 4 , n , triple square snake graph $$T\left( {C_{4,n} } \right)$$ T C 4 , n and $$m$$ m -multiple square snake graph $$M_m \left( {C_{4,n} } \right)$$ M m C 4 , n . Finally, we determine TEIS for square snake graphs, double square snake graph, triple square snake graph and $$m$$ m -multiple square snake graph, which have many applications in coding theory and physics.