JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (8): 57-62.doi: 10.6040/j.issn.1671-9352.0.2022.328

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The p-edge neighbor scattering number of graphs

Yuyuan SU(),Zongtian WEI,Yan WANG   

  1. Department of Mathematics, Xi'an University of Architecture and Technology, Xi'an 710055, Shaanxi, China
  • Received:2022-06-04 Online:2023-08-20 Published:2023-07-28

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Abstract:

In order to measure the network invulnerability more precisely, the concept of edge neighbor scattering number with edge failure probability p is combined, the concept of p-edge neighbor scattering number is proposed. The parameter calculation formulas of some basic graphs and the algorithm of p-edge neighbor scattering number of the star tree are given. The relationship between p-edge neighbor scattering number and the edge number of graphs and edge failure probability p is analyzed.

Key words: graph, edge neighbor scattering number, edge failure probability, p-edge neighbor scattering number, invulnerability

CLC Number: 

  • O157.5

Fig.1

Double star DS(m, n) of order m+n+2"

Fig.2

Comet Cn, k"

Fig.3

Generalized Petersen graph P(n, 1)"

Fig.4

Cycle star S1, C3, C4, C5, C6"

Fig.5

Star tree S1, 4, 3, 3"

Table 1

The p-edge neighbor scattering number of Pn+e"

d(u, v) ENSp(Pn+e)
2≤d(u, v)≤n-4 $ \begin{cases}1, & d(u)=1 \text { 或 } d(v)=1 ； \\ \max \left\{4 p-2 p^2, 1\right\}, & d(u) \neq 1 \text { 且 } d(v) \neq 1 \text { 。 }\end{cases}$
d(u, v)=n-3 $\begin{cases}1, & d(u)=1 \text { 或 } d(v)=1 ； \\ 4 p-2 p^2, & d(u) \neq 1 \text { 且 } d(v) \neq 1 \text { 。 }\end{cases}$
d(u, v)=n-2 2p-p2
d(u, v)=n-1 ENSp(Cn)

Table 2

Relationship between p-edge neighbor scattering number and edge failure probability p of several graphs"

Pn Cn S1, n-1 DS(m, n) Cn, k
ENSp 1 0 (n-2)p-pn-2 1+(m+n)p-pn-pm 1+kp-pk
$ \frac{\mathrm{d}}{\mathrm{d} p}\left(\mathrm{ENS}_p\right)$ 0 0 n-2-(n-2)pn-3 m+n-npn-1-mpm-1 k-kpk-1
ENS 1 0 n-3 m+n-1 k
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