Neighbor sum distinguishable $$k$$ -edge colorings of joint graphs

Abstract

In a graph G, the normal k-edge coloring \(\sigma \) is defined as the conventional edge coloring of G using the color set \(\left[ k \right] =\left\{ 1,2,\cdots ,k \right\} \). If the condition \(S\left( u \right) \ne S\left( v \right) \) holds for any edge \(uv\in E\left( G \right) \), where \(S\left( u \right) =\sum \nolimits _{uv\in E\left( G \right) }{\sigma \left( uv \right) }\), then \(\sigma \) is termed a neighbor sum distinguishable k-edge coloring of the graph G, abbreviated as k-VSDEC. The minimum number of colors \( k \) needed for this type of coloring is referred to as the neighbor sum distinguishable edge chromatic number of \( G \), represented as \( \chi '_{\varSigma }(G) \). This paper examines neighbor sum distinguishable k-edge colorings in the joint graphs of an h-order path \({{P}_{h}}\) and an \(\left( z+1 \right) \)-order star \({{S}_{z}}\), providing exact values for their neighboring and distinguishable edge coloring numbers, which are either \(\varDelta \) or \(\varDelta +1\).