Study on ( <i>r, s</i> )-generalised transformation graphs, a novel perspective based on transformation graphs

Abstract For a graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">V</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">E</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathbb{Q}}=\left({\mathbb{V}},{\mathbb{E}}) , the transformation graph are defined as graphs with vertex set being <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∪</m:mo> <m:mi mathvariant="double-struck">E</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathbb{V}}\left({\mathbb{Q}})\cup {\mathbb{E}}\left({\mathbb{Q}}) and edge set is described following certain conditions. In comparison with the structural descriptor of the original graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> </m:math> {\mathbb{Q}} , the topological descriptor of its transformation graphs displays distinct characteristics related to the structure. Thus, a compound’s transformation graph descriptors can be used to model a variety of structural features of the underlying molecular structure and initiate a structural analysis. In this work, the concept of transformation graphs is extended giving rise to a novel class of graphs, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(r,s) -generalised transformation graphs, whose vertex set is union of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> r copies of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathbb{V}}\left({\mathbb{Q}}) , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s copies of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">E</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\mathbb{E}}\left({\mathbb{Q}}) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi>N</m:mi> </m:math> r,s\in N , and the edge set are defined under certain conditions. Furthermore, this class of graphs is analysed with the help of first Zagreb index. Mainly, there are eight transformation graphs based on the criteria for edge set, but under the concept of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(r,s) -generalised transformation graphs, infinite number of graphs can be described and analysed.