An independent set degree condition for fractional critical deleted graphs

Abstract

Let i ≥ 2, ∆ ≥ 0, 1 ≤ a ≤ b − ∆, n > (a+b)(ib+2m−2) a + n and δ(G) ≥ b 2 a + n + 2m, and let g, f be two integer-valued functions defined on V (G) such that a ≤ g(x) ≤ f (x) − ∆ ≤ b − ∆ for each x ∈ V (G). In this article, it is determined that G is a fractional (g, f, n , m)-critical deleted graph if max{d 1 , d 2 , • • • , d i } ≥ b(n+n) a+b for any independent subset {x 1 , x 2 ,. .. , x i } ⊆ V (G). The result is tight on independent set degree condition.

Parameters and fractional factors in different settings

“…The concept of fractional (g, f , n , m)critical deleted graph can be regarded as a combination of fractional (g, f , n )-critical graph and fractional (g, f , m)-deleted graph, i.e., a graph G is fractional (g, f , n , m)-critical deleted if after n vertices are deleted from G, the rest of the graph is a fractional (g, f , m)-deleted graph. For some recent advances on the existence of a fractional factor in various settings, we can refer to Gao et al [7][8][9][10]12], Liu et al [18], and Zhou et al [34,35,37,[39][40][41][42][43].…”

“…The discussion below is divided into two parts based on the relationship between |S| + |T| and n. Case 1. |S| + |T| ≤ n. Let y 0 ∈ T be the vertex adjacent to the least number (this number is denoted by d) of vertices in X -S. According to (8), we have d|T| ≤ (b -)|T| -1 -|S|(a + ) and then…”

The Extension Degree Conditions for Fractional Factor

“…When functions g and f take special value for all vertices, the fractional (g, f, n ′ , m)-critical deleted graph becomes various names which are presented in Table 1. Several recent contributions in this topic were presented in Zhou et al [10], [11], [13], [15] and [16], and Gao et al [3], [4], [5] and [7], Knor et al [8], and Liu et al [9].…”

Useful Irregularity Indices in QSPR Study for Bismuth Tri-Iodide

“…In particular, graph polynomials related to graph are rich in information. Mathematical tools like polynomials and topological-based numbers [1][2][3][4][5] have significant importance to collect information about properties of chemical compounds. We can find out many hidden information about compounds through these tools.…”