The Reformulated F-Index of Vertex and Edge F-Join of Graphs
Abstract: The valency based topological indices (TIs) are defined by algebraic functions as the chemical graph theory tools in which the structural parameters are used as input and the output is related with the topology of chemical species. In theoretical chemistry, the TIs are mainly used to investigate/develop the QSAR and QSPR investigations of the molecular graphs. The Reformulated F-index or RFI is one such kind of TIs. The RFI for a molecular graph Y is defined as R … Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 4 publications
References 22 publications
Let D = V , E be an oriented graph with minimum out-degree δ + . For x ∈ V D , let d D + x and d D + + x be the out-degree and second out-degree of x in D , respectively. For a directed graph D , we say that a vertex x ∈ V D is a Seymour vertex if d D + + x ≥ d D + x . Seymour in 1990 conjectured that each oriented graph has a Seymour vertex. A directed graph D is called m -free if there are no directed cycles with length at most m in D . A directed graph D = V , E is called k -transitive if, for any directed x y -path of length k , there exists x , y ∈ E . In this paper, we show that (1) each δ + − 2 -free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in m -free and 2 m + 2 -transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).
Let D = V , E be an oriented graph with minimum out-degree δ + . For x ∈ V D , let d D + x and d D + + x be the out-degree and second out-degree of x in D , respectively. For a directed graph D , we say that a vertex x ∈ V D is a Seymour vertex if d D + + x ≥ d D + x . Seymour in 1990 conjectured that each oriented graph has a Seymour vertex. A directed graph D is called m -free if there are no directed cycles with length at most m in D . A directed graph D = V , E is called k -transitive if, for any directed x y -path of length k , there exists x , y ∈ E . In this paper, we show that (1) each δ + − 2 -free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in m -free and 2 m + 2 -transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).
On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs
Let G be a graph with V = V G . A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class { S : S is an independent set of G } and ∅ is denoted by I G . For a graph H , a function f : V ⟶ I H is called an H − independent coloring of G (or simply called an H − coloring) if f x ∩ f y = ∅ for any adjacent vertices x , y ∈ V and f V is a class of disjoint sets. Let α H , G denote the maximum cardinality of the set{ ∑ x ∈ V f x : f is an H − coloring of G }. In this paper, we obtain basic properties of an H − coloring of G and find α H , G of some families of graphs G and H . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph K n and a graph G and prove that α K n □ G = α K n , G .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
