Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 6, No. 4, October 2021 Research Paper Determining The Number of Connected Vertex Labeled Graphs of Order Seven without Loops by Observing The Patterns of Formula for Lower Order Graphs with Similar Property Muslim Ansori1, Wamiliana1*, Fadila Cahya Puri1 1Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Lampung, Bandar Lampung, 35145, Indonesia, *Corresponding author: wamiliana.1963@fmipa.unila.ac.id AbstractGiven n vertices and m edges, m ≥ 1, and for every vertex is given a label, there are lots of graphs that can be obtained. The graphsobtained may be simple or not simple, connected or disconnected. A graph G(V,E) is called simple if G(V,E) not containing loopsnor paralel edges. An edge which has the same end vertex is called a loop, and paralel edges are two or more edges which connectthe same set of vertices. Let N(G7,m,t) as the number of connected vertex labeled graphs of order seven with m vertices and t (t is the number edges that connect different pair of vertices). The result shows that N(G7,m,t) = ct C (m−1)t−1 , with c6=6727, c7=30160 , c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15= 8022, c16=2940, c17=4417, c18=2835, c19=210, c20= 21, c21=1. KeywordsGraph, Connected, Vertex, Labeled, Order, Loops Received: 7 July 2021, Accepted: 4 October 2021 https://doi.org/10.26554/sti.2021.6.4.328-336 1. INTRODUCTION Graph theory emerged as a new eld in mathematics in 1736 after Leonhard Euler gave solution to the Konigsberg problem, graph theory was used widely in many real-life applications, especially as problems representation. A Graph G (V,E) is a structure which consists of a set V={v1, v2, . . . , vn} of vertices, where V ≠ ∅, and a set of edges E=ei j | i, j ∈ V which connect the vertices of V. Usually the vertices are used to represent cities, depots, train stations, airports, etc., while edges are usu- ally used to represent roads, train tracks, ight paths, etc. A number ci j ≥ 0 can be assigned to the edge ei j as a nonformal information which can represent the distance, time, cost, ow, etc. Because the exibility of how to draw a graph, where there is no restriction in drawing an edge (can be a straight line, a curve, or other line), graph becomes an interesting structure to cope with, especially to represent the problem for easily visu- alization. Some of graph terminologies that commonly used in application is the concept of tree, where tree is a connected graph without cycle. Some applications that use graph theoretical concept as problems representation include applications in biology, chem- istry, engineering, computer science, economics, agriculture, and others. Forexample, in biology, a leaf labeled tree was used to represent the evolutionary history of a set of taxa which is called as phylogenetic tree (Huson and Bryant, 2006; Brandes and Cornelsen, 2009), and Mathur and Adlakha (2016) used combinedtree torepresentDNA, inchemistry/pharmaceutical, Gramatica et al. (2014) used graph concept to describe or rep- resent the possible modes of action for any given pharmaco- logical compound; in engineering and computer science, Hsu and Lin (2009) exposed a lot of graph theoretical concepts including Hamiltonian circuits with relation in network design, Al Etaiwi (2014) in order to generate a complex cipher text used the concepts minimum spanning tree, complete graph and cycle graph, Priyadarsini (2015) investigate the use of graph theory concept, extremal and expander graphs in de- signing some ciphers, while Ni et al. (2021) use bipartite and corona graphs to create ciphers; in economics, Álvarez and Ehnts (2015) used directed graph to represent the dynamic closures of the accounting structure; in agriculture, Kawakura and Shibasaki (2018) used graph theory concepts to group agricultural workers engaging in manual tasks, Kannimuthu et al. (2020) use graph coloring to optimize farmer’s objective, and many more. In 1857 Cayley enumerated the isomer of CnH2n+2 using the concept of tree (Cayley, 1874), and followed by Slomenski (1964) who used graph theory to calculate additive structural https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2021.6.4.328-336&domain=pdf https://doi.org/10.26554/sti.2021.6.4.328-336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 properties of hydrocarbon. Bona (2007) discussed how to enu- merate trees and forest. If we are given n vertices and m edges, then lots of graphs can be obtained using that information. The graph obtained may be simple graph which does not contain loop nor parallel edges, or maybe not simple. Moreover, the graph obtained also may be connected or disconnected. For connected vertex labeled graph, the number of graph of or- der ve with maximum number of paralel edges is ve without loopswas investigatedbyWamilianaetal. (2019), andthenum- ber of graph of order six without paralel edges with ten loops maximum also investigated by Wamiliana et al. (2020). Puri et al. (2021) investigated the number of graphs of order six with maximum thirty edges without loops. For disconnected vertex labeled graph, Wamiliana et al. (2016) investigated the number of graph of order ve without paralel edges, Amanto et al. (2017) gave the formula for graph of order maximal four, Putri et al. (2021) observed and gave formula for the number graphs of order six without loops and may contain maximum twenty parallel edges, while Pertiwi et al. (2021) proposed the formula for counting the number graph of order six without loops, especially when the graph obtained only contains maxi- mum seven loops and the number of non loop edges is even. In this study, by observing the patterns of the formula of the number of connected vertex labeled graphs of order ve and order six containing no loops, the formula of graphs of order seven with similar property will be discussed. We organized this paper as follows: Section I is Introduc- tion that describes about what is graph, some applications of graphs, and some researches related with this study. In Section II Observation and Investigation will be discussed while Result and Discussion is provided in Section III, and Conclusion in given in Section IV. 2. OBSERVATION AND INVESTIGATION Given a graph G(V,E) where n =|V| = 7 and G is connected. Because G is connected, then the number of edges m = |E| ≥ 6. Every vertex in G is labeled, therefore graphs G1 and G2 in Figure 1 are two dierent graphs even though both graphs look similar. Figure 1. Two Dierent Graphs that Look The Same but Dierent because of The Vertex Labeling Denote N(Gn,m,t) as the number of connected vertex la- beled graphs containing no loops of order n, m edges and t, where t is the number of edges that connect dierent pairs of vertices in G. Edges that connect the same pair of vertices is counted as one. Moreover, isomorphics graphs are counted as one graph. The result on Table 1 for n=5 and are obtained from Wa- miliana et al. (2019), and for n=6 from and Puri et al. (2021). From Table 1 we know that for n = 5, the maximum number of t is 10, and for n= 6 is 15, and since the graph is connected, then m ≥ 4 for n = 5, and m ≥ 5 for n =6. By observing Table 1 we found that there are patterns between those two order graphs. Notice that, for every t, the formula only dier on the coecients. Let ct is contant with t= 6,7,...,21. By using the patterns on Table 1, we predict that the formula fororder seven as ct C (m−1) t−1 . Note that for order 7, maximum t is 21. 3. RESULTS AND DISCUSSION Given n=7, t and m, the number of graphs of order seven, connected, and vertex labelled are obtained by: pattern con- struction, grouping the patterns in term of m and t, and then calculate the graphs. Starting with t=6, we construct for m ≥ 6. The process continue with t=7 until t=21 (maximum possible t for n=7). Table 2 shows some possible patterns for t=n-1. Note that we do not put all possible patterns here due to space limitation, for example, for t=n-1 and m= n, the paralel edges maybe connect vertex v1 and v2 or v4 and v5, and so on, and for t=n-1 and m=n+1, that is possible the paralel edges only on one pair of vertices, for example, there are three edges that connect vertex v1 and v2, etc. The number of graphs obtained is given in Table 3. By observing the number in every column, Table 3 can be rewrite as in Table 4. By grouping the graphs by m and t, we notice that every column of Table 4 constitute patterns. Note that in the Table 3 and 4 we are not inputting all the numbers of graph obtained because t is xed in every column and adding more edges only adding more paralel edges on t, and the pattern continues for the next m, for example: for t= 6, m ≥ 6 we only input the number until m= 12 and the pattern is 1, 6, 21, 56, 126, 252, 462 (if adding more m, the pattern becomes 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002,...). The sequence of numbers that appear in the rst column (t= 6) is 1, 6, 21, 56, 126, 252, 462 and that number is multiplied by 6727. Therefore we can claim that the value of c6 in Table 3 is 6727. © 2021 The Authors. Page 329 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 Table 1. The Formula of The Number of Connected Vertex Labeled Graph of Order N , N = 5, 6 , M Edges And T, where T is The Number of Edges that Connect Dierent Pairs of Vertices in Graph, and Containing no Loops t n 5 6 4 N (G5,m,4) = 125 × C (m−1) 3 5 N (G5,m,5) = 222 × C (m−1) 4 N (G6,m,5) = 1296 × C (m−1) 4 6 N (G5,m,6) = 205 × C (m−1) 5 N (G6,m,6) = 1980 × C (m−1) 5 7 N (G5,m,7) = 110 × C (m−1) 6 N (G6,m,7) = 3330 × C (m−1) 6 8 N (G5,m,8) = 45 × C (m−1) 7 N (G6,m,8) = 4620 × C (m−1) 7 9 N (G5,m,9) = 10 × C (m−1) 8 N (G6,m,9) = 6660 × C (m−1) 8 10 N (G5,m,10) = 1 × C (m−1) 9 N (G6,m,10) = 2640 × C (m−1) 9 11 N (G6,m,11) = 1155 × C (m−1) 10 12 N (G6,m,12) = 420 × C (m−1) 11 13 N (G6,m,13) = 150 × C (m−1) 12 14 N (G6,m,14) = 15 × C (m−1) 13 15 N (G6,m,13) = 1 × C (m−1) 14 Table 2. Some Possible Patterns for t=n-1 (n=7) Table 3. Grouping The Number of Connected Vertex Labeled Graph of Order Seven without Loops by m and t m The Number of Connected Vertex Labeled Graphs of Order Seven without Loops t 6 7 8 9 10 11 6 6727 7 40362 30160 8 141267 211120 30765 9 376712 844480 246120 21000 10 847602 2533440 1107540 189000 28364 11 1695204 6333600 3691800 945000 283640 26880 12 3107874 13933920 10152450 3465000 1560020 295680 13 27867840 24365880 10395000 6240080 1774080 14 51754560 52792740 27027000 20280260 7687680 15 105585480 63063000 56784728 26906880 16 197972775 135135000 141961820 80720640 17 270270000 324484160 215255040 18 510510000 689528840 522762240 19 1379057680 1176215040 20 2620209592 2483120640 21 4966241280 22 9481006080 © 2021 The Authors. Page 330 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 m The Number of Connected Vertex Labeled Graphs of Order Seven without Loops t 12 13 14 15 16 12 26460 13 317520 20790 14 2063880 270270 10290 15 9631440 1891890 144060 8022 16 36117900 9459450 1080450 120330 5460 17 115577280 37837800 5762400 962640 87360 18 327468960 128648520 24490200 5454960 742560 19 842063040 385945560 88164720 24547320 4455360 20 1999899720 1047566520 279188280 93279816 21162960 21 4444221600 2618916300 797680800 310932720 84651840 22 9332865360 6110804700 2093912100 932798160 296281440 23 18665730720 13443770340 5118451800 2565194940 931170240 24 35775983880 28109701620 11772439140 6555498180 2677114440 25 56219403240 25685321760 15733195632 7138971840 26 108114237000 53511087000 35757262800 17847429600 27 107022174000 77474069400 42184833600 28 206399907000 160907682600 94915875600 29 321815365200 204434193600 30 622176372720 423470829600 31 846941659200 32 1640949464700 m The Number of Connected Vertex Labeled Graphs of Order Seven without Loops t 17 18 19 20 21 17 4417 18 75089 2835 19 675801 51030 210 20 4280073 484785 3990 21 21 21400365 3231900 39900 420 1 22 89881533 16967475 279300 4410 21 23 329565621 74656890 1536150 32340 231 24 1082858469 286184745 7066290 185955 1771 25 3248575407 981204840 28265160 892584 10626 26 9023820575 3066265125 100947000 3719100 53130 27 23461933495 8858099250 328077750 13813800 230230 28 57588382215 23916867975 984233250 46621575 888030 29 134372891835 60879300300 2755853100 145044900 3108105 30 299754912555 147124975725 7265430900 420630210 10015005 31 642331955475 339519174750 18163577250 1147173300 30045015 32 1327486041315 751792458375 43313145750 2963531025 84672315 33 2654972082630 1603823911200 99001476000 7294845600 225792840 34 5153769336870 3307886816850 217803247200 17194993200 573166440 35 6615773633700 462831900300 38975317920 1391975640 36 12864004287750 952889206500 85258507950 3247943160 37 1905778413000 180547428600 7307872110 38 3711252699000 371125269900 15905368710 39 742250539800 33578000610 40 1447388552610 68923264410 41 137846528820 42 269128937220 © 2021 The Authors. Page 331 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 Table 4. Another form of Table 3 m The Number of Connected Vertex Labeled Graphs of Order Seven without Loops t 6 7 8 9 10 11 6 1 ×6727 7 6 ×6727 1×30160 8 21×6727 7×30160 1 ×30765 9 56×6727 28×30160 8 ×30765 1×21000 10 126×6727 84×30160 36 ×30765 9 ×21000 1 ×28364 11 252×6727 210×30160 120 ×30765 45 ×21000 10 ×28364 1× 26880 12 462×6727 462×30160 330 ×30765 165 ×21000 55 ×28364 11 × 26880 13 924×30160 792 ×30765 495 ×21000 220 ×28364 66 × 26880 14 1716×30160 1716 ×30765 1287 ×21000 715 ×28364 286 × 26880 15 3432 ×30765 3003×21000 2002 ×28364 1001 × 26880 16 6435×30765 6435×21000 5005×28364 3003× 26880 17 12870×21000 11440×28364 8008× 26880 18 24310×21000 24310×28364 19448× 26880 19 48620×28364 43758× 26880 20 92378×28364 92378× 26880 21 184756× 26880 22 352716× 26880 m The Number of Connected Vertex Labeled Graphs of Order Seven without Loops t 12 13 14 15 16 12 1×26460 13 12×26460 1×20790 14 78×26460 13×20790 1×10290 15 364×26460 91×20790 14×10290 1×8022 16 1365×26460 455×20790 105×10290 15×8022 1×2940 17 4368×26460 1820×20790 560×10290 120×8022 16×2940 18 12376×26460 6188×20790 2380×10290 680×8022 136×2940 19 31824×26460 18564×20790 8568×10290 3060×8022 816×2940 20 75582×26460 50388×20790 27132×10290 11628×8022 3876×2940 21 167960×26460 125970×20790 77520×10290 38760×8022 15504×2940 22 352716×26460 293930×20790 203490×10290 116280×8022 54264×2940 23 705432×26460 646646×20790 497420×10290 319770×8022 170544×2940 24 1352078×26460 1352078×20790 1144066×10290 817190×8022 490314×2940 25 2704156×20790 2496144×10290 1961256×8022 1307504×2940 26 5200300×20790 5200300×10290 4457400×8022 3268760×2940 27 10400600×10290 9657700×8022 7726160×2940 28 20058300×10290 20058300×8022 17383860×2940 29 40116600×8022 37442160×2940 30 77558760×8022 77558760×2940 31 155117520×2940 32 300540195×2940 Result 1: Given n = 7, m ≥ 6, t= 6, the number of connected graphs of order seven containing no loops is N (G7,m,6) = 6727 × C (m−1)5 . Proof: Look at the sequence of numbers above. It can be seen that from the sequence above that the xed dierence occur on the fth level. Therefore the polynomial that can represent this sequence is polynomial of order ve: © 2021 The Authors. Page 332 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 m The Number of Connected Vertex Labeled Graphs of Order Seven t 17 18 19 20 21 17 1×4417 18 17×4417 1×2835 19 153×4417 18×2835 1×210 20 969×4417 171×2835 19×210 1×21 21 4845×4417 1140×2835 190×210 20×21 1×1 22 20349×4417 5985×2835 1330×210 210×21 21×1 23 74613×4417 26334×2835 7315×210 1540×21 231×1 24 245157×4417 100947×2835 33649×210 8855×21 1771×1 25 735471×4417 346104×2835 134596×210 42504×21 10626×1 26 2042975×4417 1081575×2835 480700×210 177100×21 53130×1 27 5311735×4417 3124550×2835 1562275×210 657800×21 230230×1 28 13037895×4417 8436285×2835 4686825×210 2220075×21 888030×1 29 30421755×4417 21474180×2835 13123110×210 6906900×21 3108105×1 30 67863915×4417 51895935×2835 34597290×210 20030010×21 10015005×1 31 145422675×4417 119759850×2835 86493225×210 54627300×21 30045015×1 32 300540195×4417 265182525×2835 206253075×210 141120525×21 84672315×1 33 601080390×4417 565722720×2835 471435600×210 347373600×21 225792840×1 34 1166803110×4417 1166803110×2835 1037158320×210 818809200×21 573166440×1 35 2333606220×2835 2203961430×210 1855967520×21 1391975640×1 36 4537567650×2835 4537567650×210 4059928950×21 3247943160×1 37 9075135300×210 8597496600×21 7307872110×1 38 17672631900×210 17672631900×21 15905368710×1 39 35345263800×21 33578000610×1 40 68923264410×21 68923264410×1 41 137846528820×1 42 269128937220×1 P5(m) = a5m5 + a4m4 + a3m3 + a2m2 + a1m + a0 Substitute m = 6, 7, 8, 9, 10, 11 to the equation we get the following: 6727 = 7776a5 + 1296a4 + 216a3 + 36a2 + 6a1 + a0 (1) 40362 = 16807a5 +2401a4 +343a3 +49a2 +7a1 + a0 (2) 141267 = 32768a5 +4096a4 +512a3 +64a2 +8a1 +a0 (3) 376712 = 59049a5 +6561a4 +729a3 +81a2 +9a1 +a0 (4) 847602 = 100000a5+10000a4+1000a3+100a2+10a1+a0 (5) 1695204 = 161051a5 + 14641a4 + 1331a3 + 121a2+ 11a1 + a0 (6) Solving this system of linear equations we get a5 = 6727 120 , a4 = −100905120 , a3 = 571795 120 , a2 = − 1513575 120 , a1 = 1843198 120 and a0 = −807239120 P5(m) = a5m5 + a4m4 + a3m3 + a2m2 + a1m + a0 = 6727 120 m5 − 100905 120 m4 + 571795 120 m3 − 1513575 120 m2 + 1843198 120 m − 807239 120 = 6727 120 (m5 − 15m4 + 85m3 − 225m2 + 274m − 120) = 6727 120 (m − 1)(m − 2)(m − 3)(m − 4)(m − 5) = 6727 × (m − 1)(m − 2)(m − 3)(m − 4)(m − 5) (5 × 4 × 3 × 2 × 1) = 6727 × C (m−1)5 (7) For t=7, we can see from Table 4 that the sequence of numbers is 1, 7, 28, 84, 210, 462, 924, 1716. © 2021 The Authors. Page 333 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 Result 2: Given n = 7, m ≥ 6, t= 7, the number of connected graphs of order seven containing no loops is N (G7,m,7) = 30160 × C (m−1)6 . Proof: Look at the sequence of numbers above. It can be seen that from the sequence above that the xed dierence occur on the sixth level. Therefore the polynomial that can represent this sequence is polynomial of order six: P6(m) = a6m6 + a5m5 + a4m4 + a3m3 + a2m2 + a1m + a0 Substitute m = 7, 8, 9, 10, 11, 12, 13 to the equation we get the following: 30160 = 117649a6+16807a5+2401a4+343a3+49a2+7a1+a0 (8) 211120 = 262144a6+32768a5+4096a4+512a3+64a2+8a1+a0 (9) 844480 = 531441a6+59049a5+6561a4+729a3+81a2+9a1+a0 (10) 2533440 = 1000000a6 + 100000a5 + 10000a4 + 1000a3+ 100a2 + 10a1 + a0 (11) 6333600 = 1771561a6 + 161051a5 + 14641a4 + 1331a3+ 121a2 + 11a1 + a0 (12) 13933920 = 2985984a6 + 248832a5 + 20736a4 + 1728a3 + 144a2 + 12a1 + a0 (13) 27867840 = 4826809a6 + 371293a5 + 28561a4 + 2197a3 + 169a2 + 13a1 + a0 (14) Solving this system of linear equations we get a6 = 30160 720 , a5 = −633360720 , a4 = 5278000 720 , a3 = − 22167600 720 , a2 = 48979840 720 , a1 = −53202240720 and a0 = 21715200 720 Therefore P6(m) = a6m6 + a5m5 + a4m4 + a3m3 + a2m2 + a1m + a0 = 30160 720 m6 − 633360/720 720 m5 + 5278000 720 m4 − 22167600 720 m3 + 48979840 720 m2 − 53202240 720 m + 21715200 720 = 30160 720 (m6 − 21m5 + 175m4 − 735m3 + 1624m2 − 1764m + 720) = 30160 720 (m − 1)(m − 2)(m − 3)(m − 4)(m − 5)(m − 6) = 30160 × (m − 1)(m − 2)(m − 3)(m − 4)(m − 5)(m − 6) (6 × 5 × 4 × 3 × 2 × 1) = 30160 × C (m−1)6 (15) Doing with similar manner we get the following results: For n=7, m ≥ 8, t =8, N(G7,m,8) = 30765 × C (m−1) 7 For n=7, m ≥ 9, t =9, N(G7,m,9) = 21000 × C (m−1) 8 For n=7, m ≥ 10, t =10, N(G7,m,10) = 28364 × C (m−1) 9 For n=7, m ≥ 11, t =11, N(G7,m,11) = 26880 × C (m−1) 10 For n=7, m ≥ 12, t =12, N(G7,m,12) = 26460 × C (m−1) 11 For n=7, m ≥ 13, t =13, N(G7,m,13) = 20790 × C (m−1) 12 For n=7, m ≥ 14, t =14, N(G7,m,14) = 10290 × C (m−1) 13 For n=7, m ≥ 15, t =15, N(G7,m,15) = 8022 × C (m−1) 14 For n=7, m ≥ 16, t =16, N(G7,m,16) = 2940 × C (m−1) 15 For n=7, m ≥ 17, t =17, N(G7,m,17) = 4417 × C (m−1) 16 For n=7, m ≥ 18, t =18, N(G7,m,18) = 2835 × C (m−1) 17 For n=7, m ≥ 19, t =19, N(G7,m,19) = 210 × C (m−1) 18 For n=7, m ≥ 20, t =20, N(G7,m,20) = 21 × C (m−1) 19 For n=7, m ≥ 21, t =21, N(G7,m,21) = 1 × C (m−1) 20 Base on these result, we get Table 5. From Table 5 it can be seen that for every t, the formula consist of C (m−1)t−1 , and the dierence is on ct. © 2021 The Authors. Page 334 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 Table 5. Comparison for The Number of Connected Vertex Labeled Graphs of Order Five, Six, and Seven Containing no Loops t n 5 6 7 4 N(G5,m,4) = 125 × C (m−1) 3 5 N(G5,m,5) = 222 × C (m−1) 4 N(G6,m,5) = 1296 × C (m−1) 4 6 N(G5,m,6) = 205 × C (m−1) 5 N(G6,m,6) = 1980 × C (m−1) 5 N(G7,m,6) = 6727 × C (m−1) 5 7 N(G5,m,7) = 110 × C (m−1) 6 N(G6,m,7) = 3330 × C (m−1) 6 N(G7,m,7) = 30160 × C (m−1) 6 8 N(G5,m,8) = 45 × C (m−1) 7 N(G6,m,8) = 4620 × C (m−1) 7 N(G7,m,8) = 30765 × C (m−1) 7 9 N(G5,m,9) = 10 × C (m−1) 8 N(G6,m,9) = 6660 × C (m−1) 8 N(G7,m,9) = 21000 × C (m−1) 8 10 N(G5,m,10) = 1 × C (m−1) 9 N(G6,m,10) = 2640 × C (m−1) 9 N(G7,m,10) = 28634 × C (m−1) 9 11 N(G6,m,11) = 1155 × C (m−1) 10 N(G7,m,11) = 26880 × C (m−1) 10 12 N(G6,m,12) = 420 × C (m−1) 11 N(G7,m,12) = 26460 × C (m−1) 11 13 N(G6,m,13) = 150 × C (m−1) 12 N(G7,m,13) = 20790 × C (m−1) 12 14 N(G6,m,14) = 15 × C (m−1) 13 N(G7,m,14) = 10290 × C (m−1) 13 15 N(G6,m,13) = 1 × C (m−1) 14 N(G7,m,15) = 8022 × C (m−1) 14 16 N(G7,m,16) = 2940 × C (m−1) 15 17 N(G7,m,17) = 4417 × C (m−1) 16 18 N(G7,m,18) = 2835 × C (m−1) 17 19 N(G7,m,19) = 210 × C (m−1) 18 20 N(G7,m,20) = 21 × C (m−1) 19 21 N(G7,m,21) = 1 × C (m−1) 20 4. CONCLUSIONS From the discussion above we can conclude that the formula to count the number of connected vertex labeled graph of order seven has a similar pattern with the lower order graph with the similar property. The dierence of the formulas is on the coecient for every t. The result shows that the number of con- nected vertex labeled graphs of order seven containing no loops is N(G7,m,t)= ct C (m−1) t−1 , with c6=6727, c7= 30160 , c8=30765, c9=21000, c10=28364, c11=26880, c12=26460, c13=20790, c14=10290, c15=8022, c16=2940, c17=4417, c18=2835, c19=210, c20=21, c21=1. 5. ACKNOWLEDGEMENT This research is funded by The Research Center Universitas Lampung under Postgraduate research grant project and the athors thanks for the fund. REFERENCES Al Etaiwi, W. M. (2014). Encryption algorithm using graph theory. Journal of Scientic ResearchandReports, 3(19); 2519– 2527 Álvarez, M. C. and D. Ehnts (2015). The Roads not Taken: Graph Theory and Macroeconomic Regimes in Stock-ow Con- sistent Modeling. Levy Economics Institute of Bard College, Annandale-on-Hudson Amanto, A., W. Wamiliana, M. Usman, and R. Permatasari (2017). Counting The Number of Disconnected Vertex Labelled Graphs with Order Maximal Four. Science Interna- tional Lahore, 29(6); 1181–1186 Bona, M. (2007). Introduction to Enumerative Combinatorics. McGraw-Hill Science Brandes, U. and S. Cornelsen (2009). Phylogenetic graph models beyond trees. Discrete Applied Mathematics, 157(10); 2361–2369 Cayley, A. (1874). On the mathematical theory of isomers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47(314); 444–447 Gramatica, R., T. Di Matteo, S. Giorgetti, M. Barbiani, D. Bevec, and T. Aste (2014). Graph theory enables drug repurposing–how a mathematical model can drive the dis- covery of hidden mechanisms of action. PloS One, 9(1); 84912 Hsu, L. H. and C. K. Lin (2009). Graph Theory and Intercon- nection Networks. Taylor and Francis Group Huson, D. H. and D. Bryant (2006). Application of phyloge- netic networks in evolutionary studies. Molecular Biology and Evolution, 23(2); 254–267 Kannimuthu, S., D. Bhanu, and K. Bhuvaneshwari (2020). A novel approach for agricultural decision making using graph coloring. SN Applied Sciences, 2(1); 1–6 Kawakura, S. and R. Shibasaki (2018). Grouping Method Using Graph Theory for Agricultural Workers Engaging in Manual Tasks. Journal of Advanced Agricultural Technologies, © 2021 The Authors. Page 335 of 336 Ansori et. al. Science and Technology Indonesia, 6 (2021) 328-336 5(3); 173 –181 Mathur, R.andN.Adlakha(2016). Agraphtheoreticmodel for prediction of reticulation events and phylogenetic networks for DNA sequences. Egyptian Journal of Basic and Applied Sciences, 3(3); 263–271 Ni, B., R. Qazi, S. U. Rehman, and G. Farid (2021). Some Graph-Based Encryption Schemes. Journal of Mathematics, 2021; 1–8 Pertiwi, F., Amanto, Wamiliana, Asmiati, and Notiragayu (2021). Calculating the Number of vertices Labeled Order Six Disconnected Graphs which Contain Maximum Seven Loops and Even Number of Non-loop Edges Without Par- allel Edges. Journal of Physics: Conference Series, 1751(1); 12026 Priyadarsini, P. (2015). A survey on some applications of graphtheoryincryptography. JournalofDiscreteMathematical Sciences and Cryptography, 18(3); 209–217 Puri, F., M. Usman, M. Ansori, Y. Antoni, et al. (2021). The Formula to Count The Number of Vertices Labeled Order SixConnected Graphs with Maximum ThirtyEdges without Loops. Journal of Physics: Conference Series, 1751(1); 12023 Putri, D., Wamiliana, Fitriani, A. Faisol, and K. Dewi (2021). Determining the Number of Disconnected Vertices Labeled Graphs of Order Six with the Maximum Number Twenty Parallel Edges and Containing No Loops. Journal of Physics: Conference Series, 1751(1); 12024 Slomenski, W. (1964). Application of the Theory of Graph to Calculations of the Additive Structural Properties of Hydro- carbon. Russian Journal of Physical Chemistry, 38; 700–703 Wamiliana, A. Nuryaman, Amanto, A. Sutrisno, and N. Prayoga (2019). Determining the Number of Connected VerticesLabelledGraphofOrderFivewithMaximumNum- ber of Parallel Edges is Five and Containing No Loops. Jour- nal of Physics: Conference Series, 1338(1); 12043 Wamiliana,W.,A.Amanto, andG.T.Nagari (2016). Counting the Number of Disconnected Labeled Graphs of Order Five without Paralel Edges. Internasional Series on Interdisciplinary Research, 1(1); 4–7 Wamiliana, W., A. Amanto, M. Usman, M. Ansori, and F. C. Puri (2020). Enumerating the Number of Connected Ver- tices Labeled Graph of OrderSixwith Maximum Ten Loops and Containing No Parallel Edges. Science and Technology Indonesia, 5(4); 131–135 © 2021 The Authors. Page 336 of 336 INTRODUCTION OBSERVATION AND INVESTIGATION RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENT