Research On Graph Entropy Based On Four Domination Sets

Entropy is a measure of system uncertainty in information theory,first proposed by Shannon.As one of the methods for studying complexity measurement,entropy plays an important role and has been widely promoted and applied since it was proposed.Since Rashevsky and Trucco introduced entropy to the field of graph theory in 1955,the study of graph entropy has begun.Especially in the last two decades,graph entropy has been widely studied,and graph entropy based on various topological invariants such as degree power,distance,eigenvalue,independent sets,matching sets and so on has been proposed.In 1962,OysteinOre first proposed the concept of domination sets,and the domination theory of graphs is favored because of its unique research advantages.Therefore,the graph entropy based on domination sets,total domination sets,connected domination sets and independent domination sets is studied respectively,and the main research contents and results are as follows:1.This thesis studies the graph entropy bounds and corresponding extremal graphs based on domination sets,total domination sets,connected domination sets and independent domination sets,and obtains upper and lower bounds and corresponding extremal graphs for the connected domination entropy of tree graphs and lower bounds and corresponding extremal graphs for the independent domination entropy of connected graphs.At the same time,the entropy and other properties of the graph with different vertices are studied,and the total domination entropy properties of the double star graph under different vertices and other related properties are obtained.Finally,the entropy of different graphs is studied,and the total ranking of graph entropy of the connected domination entropy of the tree graph under the leaf node classification,the comparative ordering of the four domination entropies under the same graph class,and the comparison of the entropy of different graphs are obtained.2.This thesis studies the graph entropy of typical graphs under different graph operations,and the domination entropy of complete graphs,star graphs,comb graphs,and friendship graphs under the operations of disjoint union,joining,corona product and cartesian product was obtained,the connected domination entropy of paths,complete graphs and connected graphs under the operations of strong product,corona product and lexicograhic product,and the independent domination entropy of complete graphs,friendship graphs,spider graphs,and connected graphs under strong product,lexicograhic product and tensor product operations were obtained.