Search

In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as G O 1 (N) = ∑ u ′ v ′ ∈ E (N) (d (u ′) + d (v ′) + d (u ′) d (v ′) ) and G O 2 (N) = ∑ u ′ v ′ ∈ E (N) (d (u ′) + d (v ′)) d (u ′) d (v ′) , respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order δ and independence number ξ , the spur tree denoted as Υ δ , ξ maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique. [ABSTRACT FROM AUTHOR]

The cardinality of a largest independent set of G, denoted by α (G) , is called the independence number of G. The independent domination number i (G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by α c (G) , as the greatest integer r such that every vertex of G belongs to some independent subset X of V G with | X | ≥ r . The common independence number α c (G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality α c (G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i (G) ≤ α c (G) ≤ α (G) . In this paper, we characterize the trees T for which i (T) = α c (T) , and the block graphs G for which α c (G) = α (G) . [ABSTRACT FROM AUTHOR]

For any graph G of order p, a bijection f : V (G) → { 1 , 2 , ... , p } is called a numbering of G. The strength s t r f (G) of a numbering f of G is defined by s t r f (G) = max { f (u) + f (v) | u v ∈ E (G) } , and the strength s t r (G) of a graph G is s t r (G) = min { s t r f (G) | f   is   a   numbering   of   G }. In this paper, many open problems are solved, and the strengths of new families of graphs are determined. [ABSTRACT FROM AUTHOR]

The concept of Sombor index (S O) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index S O : S O = S O (G) = ∑ v i v j ∈ E (G) d G (v i) 2 + d G (v j) 2 , where d G (v i) is the degree of vertex v i in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work. [ABSTRACT FROM AUTHOR]

A total dominating set D of a graph G is said to be a secure total dominating set if for every vertex u ∈ V (G) \ D , there exists a vertex v ∈ D , which is adjacent to u, such that (D \ { v }) ∪ { u } is a total dominating set as well. The secure total domination number of G is the minimum cardinality among all secure total dominating sets of G. In this article, we obtain new relationships between the secure total domination number and other graph parameters: namely the independence number, the matching number and other domination parameters. Some of our results are tight bounds that improve some well-known results. [ABSTRACT FROM AUTHOR]

For positive integers s and t, the Ramsey number R (s , t) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R (3 , t) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R (3 , t) for small t's are computed, and R (3 , 35) ≥ 237 , R (3 , 36) ≥ 244 , R (3 , 37) ≥ 255 , R (3 , 38) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R (3 , t) are proposed. [ABSTRACT FROM AUTHOR]