The reflexive edge strength on some almost regular graphs

Abstract

A function f with domain and range are respectively the edge set of graph G and natural number up to , and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to 2 are called a total k -labeling where 2 . The total k -labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k -labeling, where for any edge , the weight is . The reflexive edge strength of the graph G , denoted by is the minimum k for graph G which has an edge irregular reflexive k -labelling. In this study, we obtained the of graphs which their vertex degrees show an almost regularity properties.

On H-irregular reflexive labeling of graph

“…Irregularity and weights of vertices of edges, subgraphs, etc, are the same as for total irregular labeling [6]. The smallest 𝑘 for which a reflexive (vertex or edge) 𝑘-labelling exists is known as the irregular reflexive strength of the graph [15].…”

“…The formal definition of vertex irregular reflexive 𝑘-labeling can be seen in [7,8]; they also provided some results on 𝑟𝑣𝑠(𝐺), the smallest value of 𝑘 for which such labeling exists is called the reflexive vertex strength of the graph 𝐺, where 𝐺 were prisms, wheels, fans, baskets, any graph with pendant vertex, sunlet graph, helm graph, subdivided star graph, and broom graph. While for a formal definition of edge irregular reflexive 𝑘-labeling can be seen in [8,9,10,11,12,13,14,15]. They determined the lower bound lemma and some previous results of 𝑟𝑒𝑠(𝐺), where 𝐺 were a star, double star 𝑆 𝑛,𝑛 , caterpillar graphs, generalized subdivided star, broom, double star graph 𝑆 𝑛,𝑚 , cycle, a cartesian product of cycles, join of cycle graphs and 𝐾 1 , generalized friendship graphs, wheels, prisms, basket, and fan graphs, the disjoint union of Generalized Petersen graphs.…”

On the Spatial Temporal Graph Neural Network Analysis Together with Local Vertex Irregular Reflexive Coloring for Time Series Forecasting on Passenger Density at Bus Station

“…Now, we need to update the learning weight. before that, we need take the sum of each column in the H 1 vi and divide them by the number of nodes as follows: z 1 = 0.0652, z 2 = 0.0846, z 3 = 0.0444, z 4 = 0.0227. Thus, we have z k = [0.0652 0.0846 0.0444 0.0227] where k = 1, 2, 3, 4.…”

“…The smallest number of vertex weights needed to color the vertices of G such that no two adjacent vertices share the same color is called a local vertex irregular reflexive chromatic number, denoted by χ lrvs (G). Furthermore, the minimum k required such that χ lrvs (G) = χ(G) is called a local reflexive vertex color strength, denoted by lrvcs(G) [1]- [7], [11]- [12], [17]. In Fig.…”