On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution
Abstract
:1. Introduction
- denotes the graph obtained from G by removing all elements of X;
- (resp. ) denotes the set of subtrees of G (resp. containing v);
- denotes the set of subtrees of G containing the edge ;
- denotes the weight of subtree ;
- is the sum of weights of subtrees in ;
- is the cardinality (namely the number) of the corresponding set of subtrees.
2. Subtree Generating Functions of and Wheel Graph
2.1. Subtree Generating Functions and Subtree Numbers of
- (i)
- not containing the center ,
- (ii)
- containing the center .
- is the collection of subtrees that contain neither nor ;
- is the collection of subtrees that contain , but not ;
- is the collection of subtrees that contain , but not ;
- is the collection of subtrees that contain both and .
2.2. Subtree Generating Function and Subtree Number of Wheel Graph
- is the set of subtrees of that contain neither nor ;
- is the set of subtrees of that contain , but not ;
- is the set of subtrees of that contain but not ;
- is the set of subtrees of that contain both and .
- is the set of subtrees in that contain neither nor ;
- is the set of subtrees in that contain , but not ;
- is the set of subtrees in that contain but not ;
- is the set of subtrees in that contain both and .
3. Behaviors of and in Terms of Subtrees
3.1. Subtree Numbers and
- the graph has subtrees, fewer than any other ; and
- the graph has subtrees (where and ), more than any other .
- has the second-largest subtree number; and
- has the third-largest subtree number for odd (or sufficiently large) n.
- the subtree number increment trend meets exponential growth when .
3.2. Subtree Densities of and
4. Concluding Remarks
Author Contributions
Funding
Conflicts of Interest
Appendix A. Matlab Code, Python Code, And Data
Pythoncode1. The Python code for computing the subtree number of . | |
1 | n=int (input()) |
2 | b=[−1 for i in range (n+1)] |
3 | b[0]=1 |
4 | b[1]=2 |
5 | for j in range (1,n+1): |
6 | sum1=0 |
7 | for r in range (1,j+1): |
8 | sum1=sum1+r∗b[j−r] |
9 | b[j]=b[j−1]+sum1 |
10 | file_name="n="+str(n)+".txt" |
11 | file = open (file_name,’w’) |
12 | for j in range (1,n+1): |
13 | i = n%j |
14 | s=(j+1)∗(n−i)/2+i+2∗∗(i)∗b[j]∗∗((n−i)/j) |
15 | file.write (str(s)) |
16 | file.write (’\n’) |
17 | print (s) |
MCF1a. MATLAB source code for Figure 1a. | |
1 | clear; |
2 | clc; |
3 | s1 = load(’n=33.txt’); |
4 | s2 = load(’n=58.txt’); |
5 | s3 = load(’n=75.txt’); |
6 | s4 = load(’n=80.txt’); |
7 | format long; |
8 | A =cell(1,10); |
9 | A{1,1}=’−bo’; |
10 | A{1,2}=’−go’; |
11 | A{1,3}=’−mo’; |
12 | A{1,4}=’−ro’; |
13 | subplot(2,2,1); |
14 | plot(s1,A{1,1}); |
15 | axis normal; |
16 | set(legend(’n=33’),’fontsize’,8); |
17 | xlabel(’number of non-central vertices’); |
18 | ylabel(’subtree number’); |
19 | subplot(2,2,2); |
20 | plot(s2,A{1,2}); |
21 | set(legend(’n=58’),’fontsize’,8); |
22 | xlabel(’number of non-central vertices’); |
23 | ylabel(’subtree number’); |
24 | subplot(2,2,3); |
25 | plot(s3,A{1,3}); |
26 | axis normal; |
27 | set(legend(’n=75’),’fontsize’,8); |
28 | xlabel(’number of non-central vertices’); |
29 | ylabel(’subtree number’); |
30 | subplot(2,2,4); |
31 | plot(s4,A{1,4}); |
32 | set(legend(’n=80’),’fontsize’,8); |
33 | xlabel(’number of non-central vertices’); |
34 | ylabel(’subtree number’); |
MCF1b: The MATLAB source code for Figure 1b. | |
1 | clear; |
2 | clc; |
3 | s1 = load(’n=33.txt’); |
4 | s2 = load(’n=58.txt’); |
5 | s3 = load(’n=75.txt’); |
6 | s4 = load(’n=80.txt’); |
7 | format long; |
8 | A =cell(1,10); |
9 | A{1,1}=’−bo’; |
10 | A{1,2}=’−go’; |
11 | A{1,3}=’−mo’; |
12 | A{1,4}=’−ro’; |
13 | subplot(2,2,1); |
14 | semilogy(s1,A{1,1}); |
15 | axis normal; |
16 | set(legend(’n=33’,’location’,’southeast’),’fontsize’,8); |
17 | xlabel(’number of non-central vertices’); |
18 | ylabel(’subtree number’); |
19 | subplot(2,2,2); |
20 | semilogy(s2,A{1,2}); |
21 | set(legend(’n=58’,’location’,’southeast’),’fontsize’,8); |
22 | xlabel(’number of non-central vertices’); |
23 | ylabel(’subtree number’); |
24 | subplot(2,2,3); |
25 | semilogy(s3,A{1,3}); |
26 | axis normal; |
27 | set(legend(’n=75’,’location’,’southeast’),’fontsize’,8); |
28 | xlabel(’number of non-central vertices’); |
29 | ylabel(’subtree number’); |
30 | subplot(2,2,4); |
31 | semilogy(s4,A{1,4}); |
32 | set(legend(’n=80’,’location’,’southeast’),’fontsize’,8); |
33 | xlabel(’number of non-central vertices’); |
34 | ylabel(’subtree number’); |
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n | n | n | |||
---|---|---|---|---|---|
3 | 38 | 19 | 2,899,980,984 | 35 | 269,604,917,347,967,886 |
4 | 112 | 20 | 9,128,846,611 | 36 | 848,689,059,340,934,448 |
5 | 332 | 21 | 28,736,686,630 | 37 | 2,671,587,471,512,527,895 |
6 | 1007 | 22 | 90,460,187,232 | 38 | 8,409,887,625,375,274,755 |
7 | 3110 | 23 | 284,759,535,167 | 39 | 26,473,477,146,304,448,341 |
8 | 9704 | 24 | 896,394,265,075 | 40 | 83,335,833,180,604,495,475 |
9 | 30,431 | 25 | 2,821,758,641,457 | 41 | 262,332,788,908,879,910,034 |
10 | 95,643 | 26 | 8,882,611,305,147 | 42 | 825,797,133,240,010,600,373 |
11 | 300,885 | 27 | 27,961,563,560,618 | 43 | 2,599,525,999,414,007,165,103 |
12 | 946,923 | 28 | 88,020,178,967,761 | 44 | 8,183,045,386,844,876,767,480 |
13 | 2,980,538 | 29 | 277,078,636,493,555 | 45 | 25,759,400,682,377,496,173,050 |
14 | 9,382,101 | 30 | 872,215,572,630,716 | 46 | 81,087,992,568,389,361,552,840 |
15 | 29,533,519 | 31 | 2,745,646,560,009,062 | 47 | 255,256,813,613,269,834,457,576 |
16 | 92,968,088 | 32 | 8,643,018,158,636,696 | 48 | 803,522,677,430,288,749,342,627 |
17 | 292,653,642 | 33 | 27,207,348,527,149,292 | 49 | 2,529,408,261,449,734,855,548,318 |
18 | 921,243,536 | 34 | 85,645,986,192,695,055 | 50 | 7,962,321,827,121,343,008,620,568 |
1 | 50,331,670 | 4,194,326 | 0.521736621869847 |
2 | 5,683,820,588 | 362,797,089 | 0.681159363604678 |
3 | 29,685,950,982 | 1,787,743,521 | 0.721967947748206 |
4 | 52,358,400,102 | 3,110,400,052 | 0.731884047161090 |
5 | 87,226,395,622 | 5,103,959,426 | 0.743041170007376 |
6 | 56,400,430,972 | 3,370,318,067 | 0.727584934777173 |
7 | 231,968,014,120 | 13,141,451,319 | 0.767462100378713 |
8 | 36,440,865,014 | 2,224,820,302 | 0.712140856484914 |
9 | 93,645,742,414 | 5,511,577,694 | 0.738727501280144 |
10 | 240,339,417,442 | 13,653,922,612 | 0.765314128820302 |
11 | 616,080,713,876 | 33,825,095188 | 0.791900742502907 |
12 | 9,137,267,062 | 592,843,864 | 0.670113169892776 |
13 | 14,666,890,448 | 933,106,276 | 0.683406494463958 |
14 | 23,533,982,520 | 1,468,662,129 | 0.696699813456106 |
15 | 37,748,062,639 | 2,311,599,999 | 0.709993128330647 |
16 | 60,525,953,078 | 3,638,341,646 | 0.723286440182849 |
17 | 97,015,668,926 | 5,726,566,014 | 0.736579749833999 |
18 | 155,453,553,144 | 9,013,325,743 | 0.749873057893726 |
19 | 249,013,516,065 | 14,186,519,625 | 0.763166364810336 |
20 | 398,761,664,190 | 22,328,865,632 | 0.776459670910826 |
21 | 638,376,020,719 | 35,144,507,194 | 0.789752976431952 |
22 | 1,021,684,176,864 | 55,315,680,041 | 0.803046281543951 |
3 | 116 | 0.7631578947368421 | 14 | 115,215,423 | 0.8186895664414613 |
4 | 444 | 0.7928571428571428 | 15 | 386,480,089 | 0.8178844370865525 |
5 | 1617 | 0.8117469879518072 | 16 | 1,291,505,336 | 0.8171718247840813 |
6 | 5789 | 0.8212512413108243 | 17 | 4,301,328,493 | 0.8165375712479791 |
7 | 20,519 | 0.8247186495176849 | 18 | 14,282,430,812 | 0.8159697793983819 |
8 | 72,064 | 0.8251351103783091 | 19 | 47,296,291,958 | 0.815458656779937 |
9 | 250,841 | 0.8242943051493543 | 20 | 156,239,476,051 | 0.8149961728004785 |
10 | 865,923 | 0.8230636087039588 | 21 | 514,980,557,554 | 0.8145757185906111 |
11 | 2,967,219 | 0.8218031806171793 | 22 | 1,693,994,724,188 | 0.8141918205521789 |
12 | 10,102,071 | 0.8206394655271703 | 23 | 5,561,968,202,536 | 0.813839912225434 |
13 | 34,200,012 | 0.8196030381092273 | 24 | 18,230,780,418,139 | 0.8135161559345164 |
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Yang, Y.; Wang, A.; Wang, H.; Zhao, W.-T.; Sun, D.-Q. On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution. Mathematics 2019, 7, 472. https://doi.org/10.3390/math7050472
Yang Y, Wang A, Wang H, Zhao W-T, Sun D-Q. On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution. Mathematics. 2019; 7(5):472. https://doi.org/10.3390/math7050472
Chicago/Turabian StyleYang, Yu, An Wang, Hua Wang, Wei-Ting Zhao, and Dao-Qiang Sun. 2019. "On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution" Mathematics 7, no. 5: 472. https://doi.org/10.3390/math7050472
APA StyleYang, Y., Wang, A., Wang, H., Zhao, W.-T., & Sun, D.-Q. (2019). On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution. Mathematics, 7(5), 472. https://doi.org/10.3390/math7050472