Let G be an (n, m)-graph. We say that G has property (∗) if for every pair of its adjacent vertices x and y, there exists a vertex z, such that z is not adjacent to either x or y. If the graph G has property (∗), then its complement G is connected, has diameter 2, and its Wiener index is equal to ( n 2 ) + m, i.e., the Wiener index is insensitive of any other structural details of the graph G. ...

E.Prisner in his book Graph Dynamics defines the k-path-step operator on the class of finite graphs. The k-path-step operator (for a positive integer k) is the operator S′ k which to every finite graph G assigns the graph S′ k(G) which has the same vertex set as G and in which two vertices are adjacent if and only if there exists a path of length k in G connecting them. In the paper the trees a...

Abstract A graph G is $$\alpha $$ α -excellent if every vertex of contained in some maximum independent set . In this paper, we characterize bipartite graphs, unicyclic simplicial chordal block and show that generalized Petersen -excellent.

A total global dominator coloring of a graph is proper vertex with respect to which every in dominates color class, not containing and does dominate another class. The minimum number colors required such called the chromatic number, denoted by . In this paper, trees unicyclic graphs are explored.

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be graph. Then, the of G is defined as SO(G)=?uv?E(G)dG2(u)+dG2(v). In this paper, we give some lemmas that can used to compare indices between two graphs. With these lemmas, determine graph with maximum SO among all cacti n vertices and k cut edges. Furthermore, unique p pendant...

Abstract Many chemical indices have been invented in theoretical chemistry, such as Wiener index, Merrifield-Simmons index, Hosoya index, spectral radius and Randić index, etc. The extremal trees and unicyclic graphs for these chemical indices are interested in existing literature. Let G be a molecular graph (called a cacti), which all of blocks of G are either edges or cycles. Denote G (n, r) ...

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of in polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute second graded Betti numbers JG, obtain minimal presentation it when is tree or unicyclic graph. We classify all graphs whose ideals are almost complete intersection, prove that they generated by d-sequence Rees algebra their Cohen-Macaulay. also an ...