Bounds On The Vertex Co-PI Index Of Trees And Unicyclic Graphs

In chemical theory, the topological index can be used to understand the physical and chemical properties of compound. Different indices reflect different properties of the molecules. The study of molecular topological index and invariant of molecular graphs is one of the research fields of chemical graph. Every vertex of simple undirected graph G=(V,E) represents an atom, every edge represents chemical bonds between atoms, this graph is known as a molecular graph. In a molecular graph, we put the vertex number and edge number as a stable invariant in the molecules. In practically, different values describe the different measurable molecular chemical and physical properties, so to connect the topological properties with measurable physical and chemical properties of the molecule, it is necessary to introduce some numeric quantity, which are associated with certain properties of molecular graph. Molecular topological indices have important application in physics, chemistry, biology and many other subjects.Let G=(V,E) be a connected graph. For e=uv∈E(G),let nu(e) be the number of vertices closer to u than to v, nv(e) be the number of vertices closer to v than to u. The vertex PI index of G is defined as Similar to the vertex PI index, the vertex Co-PI index of G is defined as Co-PIv(G)=∑|nu(e)-nv(e))|. There are a lot of researches on the PI and vertex PI indices, and there are only some preliminary researches on vertex Co-PI index. In this thesis, we obtain that Pn has the minimum vertex Co-PI index and Sn has the maximum vertex Co-PI index in trees. We also obtain that the cycle Cn has minimum vertex Co-PI index in unicyclic graphs. When4≤n≤8, the unicyclic graph Un,3w1(Sn-2)(a triangle with a star on one of its end vertices) has the maximum vertex Co-PI index. When n≥8, the unicyclic graph Un,4w1(Sn-3)(a quadrangle with a star on one of its end vertices) has the maximum vertex Co-PI index. Moreover, the second, third, fourth minimum and the second maximum values of this index for unicyclic graphs are given. Finally, the third and fourth maximum values of this index for unicyclic graphs are given in the appendix.