Iranian Journal of Mathematical Chemistry
Volume:13 Issue: 4, Autumn 2022

We obtain some sharp bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees with given order and diameter 3, graphs with given order and domination number 2, and for the join of two graphs with given order and number of vertices having maximum possible degree. Extremal graphs are presented for all the bounds.

Quantitatively, the equilibrium in classical thermodynamics in the C-component isobaric-isothermal system is determined by the minimum value of the Gibbs function. The topological manifold of this function is a 2-D dimensional, smooth piece, geometric creation. These pieces represent individual states of single-phase systems. Successive pieces of the manifold are glued along the line of phase transitions to form the manifold of the whole, en bloc, C-component system. Gluing smooth pieces together must guarantee the continuity of the glued whole. The study found the dependence of the number of ways of gluing single-phase pieces on the number of components of the system. It has also been shown that the distribution of components in individual phases of the system is represented by a planar graph with 4 faces, called a normal graph.Studies of the topological properties of the manifold fragments representing single-phase equilibrium states indicate that the value of the Gibbs potential in these states is encoded in the geometry of the topological manifold. In concrete terms, this value is equal to the length of the minimum path lying on the surface of the manifold, connecting the various degrees of freedom of the system (the vertices of the graph). In complex systems, with very large C, the number of paths connecting the degrees of freedom is monstrously large. Preliminary calculations show that in such systems the number of paths with a minimum length or not much different from it may be greater than one.

Let G=(V, E) be a simple graph with vertex set V and edge set E. The Sombor index of the graph G is a degree-based topological index, defined as SO(G)= ∑uv∈E √(d(u)2+d(v)2), in which d(x) is the degree of the vertex x∈V for x=u, v. In this paper, we characterize the extremal trees with given degree sequence that minimizes and maximizes the Sombor index.

We introduce the concept of color matrix and color energy of semigraphs. The color energy is the sum of the absolute values of the eigenvalues of the color matrix. Some properties and bounds on color energy of semigraphs are established.

The degree and distance both are significant conceptsin graphs with widespread utilization. The combined study ofthese concepts has given a new direction to the topological in-dices. In this article, we present the generalized degree distanceindices (Generalized First Schultz indices) DD(a;b), and generalizedGutman indices (Second Schultz indices) ZZ(a;b). The computedvalues of these indices on certain families of graphs along with somebounds and characterizations are obtained. Also, we present therelationship between DD(a;b) and ZZ(a;b). Further, we present theSchultz polynomials along with the statistical analysis of certaingraphs.