Among the huge diversity of ideas that show up while studying graph theory,one that has obtained a lot of popularity is the concept of labelings of graphs.Graph labelings give valuable mathematical models for a wide s...Among the huge diversity of ideas that show up while studying graph theory,one that has obtained a lot of popularity is the concept of labelings of graphs.Graph labelings give valuable mathematical models for a wide scope of applications in high technologies(cryptography,astronomy,data security,various coding theory problems,communication networks,etc.).A labeling or a valuation of a graph is any mapping that sends a certain set of graph elements to a certain set of numbers subject to certain conditions.Graph labeling is a mapping of elements of the graph,i.e.,vertex and for edges to a set of numbers(usually positive integers),called labels.If the domain is the vertex-set or the edge-set,the labelings are called vertex labelings or edge labelings respectively.Similarly,if the domain is V(G)[E(G)],then the labeling is called total labeling.A reflexive edge irregular k-labeling of graph introduced by Tanna et al.:A total labeling of graph such that for any two different edges ab and a'b'of the graph their weights has wt_(x)(ab)=x(a)+x(ab)+x(b) and wt_(x)(a'b')=x(a')+x(a'b')+x(b') are distinct.The smallest value of k for which such labeling exist is called the reflexive edge strength of the graph and is denoted by res(G).In this paper we have found the exact value of the reflexive edge irregularity strength of the categorical product of two paths (P_(a)×P_(b))for any choice of a≥3 and b≥3.展开更多
文摘Among the huge diversity of ideas that show up while studying graph theory,one that has obtained a lot of popularity is the concept of labelings of graphs.Graph labelings give valuable mathematical models for a wide scope of applications in high technologies(cryptography,astronomy,data security,various coding theory problems,communication networks,etc.).A labeling or a valuation of a graph is any mapping that sends a certain set of graph elements to a certain set of numbers subject to certain conditions.Graph labeling is a mapping of elements of the graph,i.e.,vertex and for edges to a set of numbers(usually positive integers),called labels.If the domain is the vertex-set or the edge-set,the labelings are called vertex labelings or edge labelings respectively.Similarly,if the domain is V(G)[E(G)],then the labeling is called total labeling.A reflexive edge irregular k-labeling of graph introduced by Tanna et al.:A total labeling of graph such that for any two different edges ab and a'b'of the graph their weights has wt_(x)(ab)=x(a)+x(ab)+x(b) and wt_(x)(a'b')=x(a')+x(a'b')+x(b') are distinct.The smallest value of k for which such labeling exist is called the reflexive edge strength of the graph and is denoted by res(G).In this paper we have found the exact value of the reflexive edge irregularity strength of the categorical product of two paths (P_(a)×P_(b))for any choice of a≥3 and b≥3.
