Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 7, No. 3, July 2022 Research Paper Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges Muslim Ansori1, Wamiliana1*, Fitriani1, Yudi Antoni1, Desiana Putri1 1Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Lampung, Bandarlampung, 35145, Indonesia *Corresponding author: wamiliana.1963@fmipa.unila.ac.id AbstractA graph that is connected G(V,E) is a graph in which there is at least one path connecting every two vertices in G; otherwise, it is calleda disconnected graph. Labels or values can be assigned to the vertices or edges of a graph. A vertex-labeled graph is one in whichonly the vertices are labeled, and an edges-labeled graph is one in which only edges are assigned values or labels. If both verticesand edges are labeled, the graph is referred to as total labeling. If given n vertices and m edges, numerous graphs can be made,either connected or disconnected. This study will be discussed the number of disconnected vertices labeled graphs of order sevencontaining no parallel edges and may contain loops. The results show that number of vertices labeled connected graph of order seven with no parallel edges is N(G7,m,g)l= 6,727×Cm6 ; while for 7≤g≤ 21, N(G7,m,g)l= kg C(m−(g−6))g−1 , where k7 =30,160, k8 = 30,765, k9=21,000, k10 =28,364, k11= 26,880, k12 =26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 =210, k20 = 21, k21 = 1. KeywordsVertex Labeled, Connected Graph, Order Seven, Loops Received: 5 April 2022, Accepted: 13 July 2022 https://doi.org/10.26554/sti.2022.7.3.392-399 1. INTRODUCTION Without any doubt, one of the most widely used elds of ma- thematics is graph theory, especially to represent a real-life problem because of the exibility of drawing a graph. The date of birth of graph theory began with the publication of Euler’s solution regarding the Konigsberg problem in 1736, one of the mathematical elds with specic birth date is graph theory (Vasudev, 2006). Given a set V={v1, v2, · · · , vn} of vertices/nodes, V≠∅, and a set of edges E={ei j|i,j𝜖V}, a graph G(V,E) is a structure that consists of ordered pair of V and E. In real-life problems, the cities, buildings, airports, and others can be portrayed by vertices, while the roads that connect the cities, the pipes that connect the buildings, the ight paths that connect the airports, and others can be portrayed by edges. In order to represent real-life problems, on the edges, we can assign nonstructural information such as distance, time, ow, cost, and others by setting a non-negative number ci j≥0. There is a lot of application of graph theory in real-life problems, such as in computer science, chemistry, biology, so- ciology, agriculture, and others. In computer science, the graph structure plays an important role, for example, in designing a database, software engineering, and network system (Singh, 2014). The database is used for interconnecting analysis, as a storage system with index-free adjacency, as a tool for graph- like-query, and for other purposes. In network design, graph- based representation makes the problem easier to visualize and provides a more accurate denition. In agriculture, the dynamic closures of the accounting structure are represented by a directed graph (Álvarez and Ehnts, 2015). The structure of graphs, together with discrete mathematics, are applied in chemistry to model the biological and physical properties of chemical compounds (Burch, 2019). The theoretical graph concept also was used by Gramatica et al. (2014) to repre- sent or describe the possible modes of action of a given phar- macological compound. In biology, a phylogenetic tree was represented by a leaf-labeled tree (Huson and Bryant, 2006; Brandes and Cornelsen, 2009), while Mathur and Adlakha (2016) represented DNA using a combined tree. Hsu and Lin (2008) presented many graph theoretical concepts in en- gineering and computer science, and Al Etaiwi (2014) used the concepts of a complete graph, cycle graph, and minimum spanning tree to generate a complex cipher text. Priyadarsini (2015) explored the use of graph theory concepts, expander, and extremal graphs, in the design of some ciphers, whereas Ni et al. (2021) created ciphers using corona and bipartite graphs. In agriculture, graph theory concepts were used to https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2022.7.3.392-399&domain=pdf https://doi.org/10.26554/sti.2022.7.3.392-399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 group agricultural workers performing manual tasks (Kawakura and Shibasaki, 2018), while the concept of graph coloring to optimize a farmer’s goal was used by Kannimuthu et al. (2020). The relationship and unication of graph theory and physical- chemical measures (such as boiling and melting point, covalent and ionic potentials, and electronic density) make molecular topology can describe molecular structure comprehensively. A weighted directed graph, connectivity matrix, and Dijkstra’s al- gorithm were used by Holmes et al. (2021) in plasma chemical reaction engineering. The basic structure of a directed graph is mostly used for the visualization of the reactions. Moreover, they use Gelphi, an open-source graph software for visualiza- tion. In 1874, Cayley counted the number of hydrocarbon iso- mers CnH2n+2 (Cayley, 1874), and this process is similar to enumerating the number of a binary tree. Bona (2007) dis- cussed the method of enumerating trees and forests. Redeld and Pólya are two other mathematicians that worked indepen- dently with graphical enumeration, especially in graph coloring (Bogart, 2004), and in graph enumeration, a comprehensive explanation of Pólya’s counting theorem is one of the most powerful tools. The number of graphs that can be formed for labeled and unlabeled graphs is dierent if we are given n vertices and m edges. For example, given n= 3 and m= 2, the number of sim- ple connected unlabeled graphs that can be constructed is only one, while if every vertex is assigned labeled, The maximum number of graphs that can be created is three. The higher the order of the graph, the more labeled graph are formed. Ag- narsson and Greenlaw (2006) gave the formula to enumerate graphs. However, no formula for enumerating graphs with spe- cial properties such as planarity or connectivity was provided. There are some studies that have been done concerning the enumeration of the vertex-labeled graph with connectivity properties. In 2017, Amanto et al. (2017) proposed the for- mula to count disconnected vertices labeled graphs of order maximal four. For order ve, the number of labeled vertices in connected graphs with no loops and may contain maximal ve parallel edges had been proposed by Wamiliana et al. (2019). Amanto et al. (2021) studied the relationship between the formula for the number of connected vertex labels with no loops in graphs of order ve and order six. Wamiliana et al. (2020) also discussed the number of vertices labeled connected graphs of order six with no parallel edges and a maximum of ten loops, while Puri et al. (2021) gave the formula to compute the number of vertices labeled connected graphs of order six without loops, while Ansori et al. (2021) proposed the number of vertices labeled connected graphs of order seven with no loops. The article is organized as follows: Section I provides infor- mation about graphs, graph applications in various elds, and previous research related to this topic. Section II discusses Ob- servation and Investigation, while Section III discusses Results and Discussion. Section IV contains the conclusion. 2. OBSERVATION AND INVESTIGATION Suppose that we are given the number of vertices n= 7, and the number of edges m. We will construct connected graphs G(V,E) of order n. Since the graph must be connected, then m≥6. Moreover, every vertex is labeled. Let g as the number of non-loop edges, g≥n–1. We start rstly by constructing all basic patterns of con- nected graphs of order seven. Note that the basic patterns contain no loops. The basic pattern starts with m= 6, and with n= 7, m= 6, and constructs all possible patterns. After all possi- ble patterns for m= 6 are already constructed, then we continue with m= 7, and so on until m= 21. When m= 21, only one pat- tern can be constructed because parallel edges are not allowed. Figure 1 shows some examples of patterns for m= 6, Figure 2 shows some patterns that are isomorphic with the rst graph in the second row of the graphs in Figure 1, and Figure 3 shows when m= 21. Figure 1. Some Basic Patterns for n= 7 and m= 6 Note that all isomorphic graphs will be counted in the pat- tern. However, we do not need to construct isomorphic graphs. Figure 2 shows the patterns of isomorphic graphs of the pattern of the rst graph in the second row of Figure 1. Figure 2. Some Patterns are Isomorphic with the First Graph in the Second Row in Figure 1 Figure 3. The Basic Pattern for n= 7 and m= 21 © 2022 The Authors. Page 393 of 399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 Figure 4. The Procedure After constructing the basic pattern, the enumeration step begins. It begins from the rst pattern of n= 7 dan m= 6 by adding one loop so that m= 7, calculating the number of graphs thatareable tobeformed, andthencontinuingwith thispattern by increasing the number of loops (m= 9), and so on. Continue with this similar manner until the last pattern. The procedure can be put in the following diagram: 3. RESULTS AND DISCUSSION The rst step, as given in Figure 4 is constructing all possible patterns. Because there are many patterns obtained and due to limitation of space, here we give some patterns and also the number of all possible graphs formed according to the patterns. The obtained graphs are grouped by m and g, for example, for n=6, m= 6, and g= 6, the patterns are: The results for all patterns are shown in Table 1 below: Please note that in the table the dash sign (−) means there is impossible to construct the graph, while the empty space on the table means that we are not calculate more because g is xed in each column, adding more edges simply adds more loops, and the constructed graph already constitute a sequence of numbers. The number in each column is able to be written as multiplication of a x number and a sequence of number so that Table 2 can be rewritten in Table 3 as follow: From Table 3 we can see that for every g= 6, 7, · · · , 21, the number of graphs obtained are bigger as m increases, and the number of graphs obtained are multiplication of a x number. For example, for g= 6, the x number is 6,727, and the number of graphs increases follows a certain pattens of sequence which is 1, 7, 28, 84, 210, 462, 924, 1,716, 3,003, 5,005. 1 7 28 84 210 462 924 1716 3003 5005 6 21 56 126 252 462 792 1287 2002 15 35 70 126 210 330 495 715 Table 1. The Pattern for n= 7, m= 6, and g= 6 The patterns The number of isomorphic graphs C71.C 6 6= 7 7! 2 = 2,520 C71.C 6 3.C 3 2= 420 C71.C 6 2.C 4 3.1= 420 C71.C 6 3.3!= 840 C71.C 6 1.C 5 2.C 3 3= 420 C71.C 6 1.C 5 1.C 4 4= 210 C72.C 5 1.C 4 2.2!= 1,260 C72.C 5 1.C 4 2.C 2 2= 630 Total 6,727 © 2022 The Authors. Page 394 of 399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 Table 2. The Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges The number of vertices labeled connected order seven graphs with no parallel edges m g 6 7 8 9 10 11 6 6,727 − − − − − 7 47,089 30,160 − − − − 8 188,356 211,120 30,765 − − − 9 565,068 844,480 215,355 21,000 − − 10 1,412,670 2,533,440 861,420 147,000 28,364 − 11 3,107,874 6,333,600 2,584,260 588,000 198,548 26,880 12 6,215,748 13,933,920 6,460,650 1,764,000 794,192 188,160 13 11,543,532 27,867,840 14,213,430 4,410,000 2,382,576 752,640 14 20,201,181 51,754,560 28,426,860 9,702,000 5,956,440 2,257,920 15 33,668,635 90,570,480 52,792,740 19,404,000 13,104,168 5,644,800 16 − 150,950,800 92,387,295 36,036,000 26,208,336 12,418,560 17 − − 153,978,825 63,063,000 48,672,624 24,837,120 18 − − − 105,105,000 85,177,092 46,126,080 19 − − − − 141,961,820 80,720,640 20 − − − − − 134,534,400 The number of vertices labeled connected order seven graphs with no parallel edges m g 12 13 14 15 16 12 26,460 − − − − 13 185,220 20,790 − − − 14 740,880 145,530 10,290 − − 15 2,222,640 582,120 72,030 8,022 − 16 5,556,600 1,746,360 288,120 56,154 2,940 17 12,224,520 4,365,900 864,360 224,616 20,580 18 24,449,040 9,604,980 2,160,900 673,848 82,320 19 45,405,360 19,209,960 4,753,980 1,684,620 246,960 20 79,459,380 35,675,640 9,507,960 3,706,164 617,400 21 132,432,300 62,432,370 17,657,640 7,412,328 1,358,280 22 − 104,053,950 30,900,870 13,765,752 2,716,560 23 − − 51,501,450 24,090,066 5,045,040 24 − − − 40,150,110 8,828,820 25 − − − − 14,714,700 The number of vertices labeled connected order seven graphs with no parallel edges m g 12 13 14 15 16 17 4,417 − − − − 18 30,919 2,835 − − − 19 123,676 19,845 210 − − 20 371,028 79,380 1,470 21 − 21 927,570 238,140 5,880 147 1 22 2,040,654 595,350 17,640 588 7 23 4,081,308 1,309,770 44,100 1,764 28 24 7,579,572 2,619,540 97,020 4,410 84 25 13,264,251 4,864,860 194,040 97,02 210 26 22,107,085 8,513,505 360,360 19,404 462 27 − 14,189,175 630,630 36,036 924 28 − − 1,051,050 63,063 1,716 29 − − − 105,105 3,003 30 − − − − 5,005 © 2022 The Authors. Page 395 of 399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 Table 3. Alternative form of Table 2 The number of vertices labeled connected order seven graphs with no parallel edges m g 6 7 8 9 10 11 6 1x6,727 − − − − − 7 7x6,727 1x30,160 − − − − 8 28x6,727 7x30,160 1x30,765 − − − 9 84x6,727 28x30,160 7x30,765 1x21,000 − − 10 210x6,727 84x30,160 28x30,765 7x21,000 1x28,364 − 11 462x6,727 210x30,160 84x30,765 28x21,000 7x28,364 1x26,880 12 924x6,727 462x30,160 210x30,765 84x21,000 28x28,364 7x26,880 13 1,716x6,727 924x30,160 462x30,765 210x21,000 84x28,364 28x26,880 14 3,003x6,727 1,716x30,160 924x30,765 462x21,000 210x28,364 84x26,880 15 5,005x6,727 3,003x30,160 1,716x30,765 924x21,000 462x28,364 210x26,880 16 − 5,005x30,160 3,003x30,765 1,716x21,000 924x28,364 462x26,880 17 − − 5,005x30,765 3,003x21,000 1,716x28,364 924x26,880 18 − − − 5,005x21,000 3,003x28,364 1,716x26,880 19 − − − − 5,005x28,364 3,003x26,880 20 − − − − − 5,005x26,880 The number of vertices labeled connected order seven graphs with no parallel edges m g 12 13 14 15 16 12 1x26,460 − − − − 13 7x26,460 1x20,790 − − − 14 28x26,460 7x20,790 1x10,290 − − 15 84x26,460 28x20,790 7x10,290 1x8022 − 16 210x26,460 84x20,790 18x10,290 7x8,022 1x2,940 17 462x26,460 210x20,790 84x10,290 28x8,022 7x2,940 18 924x26,460 462x20,790 210x10,290 84x8,022 28x2,940 19 1,716x26,460 924x20,790 462x10,290 210x8,022 84x2,940 20 3,003x26,460 1,716x20,790 924x10,290 462x8,022 210x2,940 21 5,005x26,460 3,003x20,790 1,716x10,290 924x8,022 462x2,940 22 − 5,005x20,790 3,003x10,290 1,716x8,022 924x2,940 23 − − 5,005x10,290 3,003x8,022 1,716x2,940 24 − − − 5,005x8,022 3,003x2,940 25 − − − − 5,005x2,940 The number of vertices labeled connected order seven graphs with no parallel edges m g 12 13 14 15 16 17 1x4,417 − − − − 18 7x4,417 1x2,835 − − − 19 28x4,417 7x2,835 1x210 − − 20 84x4,417 28x2,835 7x210 1x21 − 21 210x4,417 84x2,835 28x210 7x21 1x1 22 462x4,417 210x2,835 84x210 28x21 7x1 23 924x4,417 462x2,835 210x210 84x21 28x1 24 1,716x4,417 924x2,835 462x210 210x21 84x1 25 3,003x4,417 1,716x2,835 924x210 462x21 210x1 26 5,005x4,417 3,003x2,835 1,716x210 924x21 462x1 27 − 5,005x2,835 3,003x210 1,716x21 924x1 28 − − 5,005x210 3,003x21 1,716x1 29 − − − 5,005x21 3,003x1 30 − − − − 5,005x1 © 2022 The Authors. Page 396 of 399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 20 35 56 84 120 165 220 15 21 28 36 45 55 6 7 8 9 10 1 1 1 1 Notate N(G7,m,g)l as the number of vertices labeled con- nectedgraphsofordersevencontainingnoparallel edges (loops are allowable) with the number of edges is m and the number of non loop edges is g. Result 1: Given m≥6, g= 6, the total number of vertices labeled connected graphs of order seven with no parallel edges is N(G7,m,g)l= 6,727×Cm6 Proof: Consider the above sequence of numbers. That sequence of numbers is able to be represented by polyno- mial of order six because the xed Q5m = 𝛼6m 6 + 𝛼5m5 + 𝛼4m4 + 𝛼3m3 + 𝛼2m2 + 𝛼1m + 𝛼0 The following system of equations is obtained by substituting m= 6, 7, 8, 9, 10, 11, 12 to Q5(m). 6, 727 = 46, 656𝛼6 + 7, 776𝛼5 + 1, 296𝛼4 + 216𝛼3 + 36𝛼2 + 6𝛼1 + 𝛼0 (1) 47, 089 = 117, 649𝛼6 + 16, 807𝛼5 + 2, 401𝛼4 + 343𝛼3 + 49𝛼2 + 7𝛼1 + 𝛼0 (2) 188, 356 = 262, 144𝛼6 + 32, 768𝛼5 + 4, 096𝛼4 + 512𝛼3 + 64𝛼2 + 8𝛼1 + 𝛼0 (3) 565, 068 = 531, 441𝛼6 + 59, 049𝛼5 + 6, 561𝛼4 + 729𝛼3 + 81𝛼2 + 9𝛼1 + 𝛼0 (4) 1, 412, 670 = 1, 000, 000𝛼6 + 100, 000𝛼5 + 10, 000𝛼4 + 1, 000𝛼3 + 100𝛼2 + 10𝛼1 + 𝛼0 (5) 3, 107, 874 = 1, 771, 561𝛼6 + 161, 051𝛼5 + 14, 641𝛼4 + 1, 331𝛼3 + 121𝛼2 + 11𝛼1 + 𝛼0 (6) 6, 215, 748 = 2, 985, 984𝛼6 + 248, 832𝛼5 + 20, 736𝛼4 + 1, 728𝛼3 + 144𝛼2 + 12𝛼1 + 𝛼0 (7) These equations form a system of equations that can be trans- formed into a matrix Ax= b as follow:  46, 656 7, 776 1, 296 216 36 6 1 117, 649 16, 807 2, 401 343 49 7 1 262, 144 32, 768 4, 096 512 64 8 1 531, 441 59, 049 6, 561 729 81 9 1 1, 000, 000 100, 000 10, 000 1, 000 100 10 1 1, 771, 561 161, 051 14, 641 1, 331 121 11 1 2, 985, 984 248, 832 20, 736 1, 728 144 12 1   𝛼6 𝛼5 𝛼4 𝛼3 𝛼2 𝛼1 𝛼0  =  6, 727 47, 089 188, 356 565, 068 1, 412, 670 3, 107, 874 6, 215, 748  By solving this system of equations we get: 𝛼6= 6,727 720 , 𝛼5= −100,905720 , 𝛼4= 57,175 720 , 𝛼3= − 151,575 720 , 𝛼2= 1,843,198 720 , 𝛼1= −807,240720 , 𝛼0= 0. Thus Q5(m) =𝛼6m6 + 𝛼5m5 + 𝛼4m4 + 𝛼3m3 + 𝛼2m2 + 𝛼1m+𝛼0 = 6, 727 720 m6 − 100, 905 720 m5 + 57, 175 720 m4 − 151, 575 720 m3 + 1, 843, 198 720 m2 − 807, 240 720 m = 6, 727 720 (m6 − 15m5 + 85m4 − 225m3 + 274m2 − 120m) = 6, 727(m − 1) (m − 2) (m − 3) (m − 4) (m − 5) (m − 6)! 6.5.4.3.2.1(m − 6)! =6, 727 × Cm6 Please note that for every g (every column, the sequence of numbers is the same, except the multiplier). Thus, the polynomial related to the sequence of numbers is the same. However, because the multipliers are dierent, it will cause dierent formulas. Result 2: Given m≥7, g= 7, the total number of vertices labeled connected graphs of order seven with no parallel edges is N(G7,m,g)l= 30,160×C (m−1) 6 Proof: The polynomial that represents the sequence is the same which is Q5m = 𝛼6m 6 + 𝛼5m5 + 𝛼4m4 + 𝛼3m3 + 𝛼2m2 + 𝛼1m + 𝛼0 However, for m= 7, the numbers of graphs are dierent. The following system of equations is obtained by substituting m= 7, 8, 9, 10, 11, 12, 13 to the equation. 30, 160 = 117, 649𝛼6 + 16, 807𝛼5 + 2, 401𝛼4 + 343𝛼3 + 49𝛼2 + 7𝛼1 + 𝛼0 (8) 211, 120 = 262, 144𝛼6 + 32, 768𝛼5 + 4, 096𝛼4 + 512𝛼3 + 64𝛼2 + 8𝛼1 + 𝛼0 (9) 844, 480 = 531, 441𝛼6 + 59, 049𝛼5 + 6, 561𝛼4 + 729𝛼3 + 81𝛼2 + 9𝛼1 + 𝛼0 (10) 2, 533, 440 = 1, 000, 000𝛼6 + 100, 000𝛼5 + 10, 000𝛼4 + 1, 000𝛼3 + 100𝛼2 + 10𝛼1 + 𝛼0 (11) 6, 333, 600 = 1, 771, 561𝛼6 + 161, 051𝛼5 + 14, 641𝛼4 + 1, 331𝛼3 + 121𝛼2 + 11𝛼1 + 𝛼0 (12) 13, 933, 920 = 2, 985, 984𝛼6 + 248, 832𝛼5 + 20, 736𝛼4 + 1, 728𝛼3 + 144𝛼2 + 12𝛼1 + 𝛼0 (13) 27, 867, 840 = 4, 826, 809𝛼6 + 371, 293𝛼5 + 28, 561𝛼4 + 2, 197𝛼3 + 169𝛼2 + 13𝛼1 + 𝛼0 (14) These equations form a system of equations that can be trans- formed into a matrix Ax= b as follow: 117, 649 16, 80 2401 343 49 7 1 262, 144 32, 768 4096 512 64 8 1 531, 441 59, 049 6561 729 81 9 1 1, 000, 000 100, 000 10, 000 1000 100 10 1 1, 771, 561 161, 051 14, 641 1331 121 11 1 2, 985, 984 248, 832 20, 736 1728 144 12 1 4, 826, 809 371, 293 28, 561 2197 169 13 1   𝛼6 𝛼5 𝛼4 𝛼3 𝛼2 𝛼1 𝛼0  =  30, 160 211, 120 844, 480 2, 533, 440 6, 333, 600 13, 933, 920 27, 867, 840  By solving this system of equations we get: 𝛼6= 30,160 720 , 𝛼5= −633,360720 , 𝛼4= 5,278,000 720 , 𝛼3= − 22,167,600 720 , 𝛼2= 48,979,840 720 , 𝛼1= −53,202,240720 , and 𝛼0= 21,715,200 720 . Thus Q5(m) =𝛼6m6 + 𝛼5m5 + 𝛼4m4 + 𝛼3m3 + 𝛼2m2 + 𝛼1m+𝛼0 = 30, 160 720 m6 − 633, 360 720 m5 + 5, 278, 000 720 m4 − 22, 167, 600 720 m3 + 48, 979, 840 720 m2 − 53, 202, 240 720 m + 21, 715, 200 720 = 30, 160 720 (m6 − 21m5 + 175m4 − 735m3 + 624m2 − 1764m + 720) = 30, 160 720 (m − 1) (m − 2) (m − 3) (m − 4) (m − 5) (m − 6) = 30, 160(m − 1) (m − 2) (m − 3) (m − 4) (m − 5) (m − 6) (m − 7)! 6.5.4.3.2.1(m − 7)! =30, 160 × C (m−1)6 The following results are obtained by doing the similar manner: For m≥8, g=8, is N(G7,m,g)l = 30,765×C (m−2) 7 For m≥9, g=9, is N(G7,m,g)l = 21,000×C (m−3) 8 For m≥10, g=10, is N(G7,m,g)l = 28,364×C (m−4) 9 For m≥11, g=11, is N(G7,m,g)l = 26,880×C (m−5) 10 For m≥12, g=12, is N(G7,m,g)l = 26,460×C (m−6) 11 For m≥13, g=13, is N(G7,m,g)l = 20,790×C (m−7) 12 For m≥14, g=14, is N(G7,m,g)l = 10,290×C (m−8) 13 © 2022 The Authors. Page 397 of 399 Ansori et. al. Science and Technology Indonesia, 7 (2022) 392-399 For m≥15, g=15, is N(G7,m,g)l = 8,022×C (m−9) 14 For m≥16, g=16, is N(G7,m,g)l = 2,940×C (m−10) 15 For m≥17, g=17, is N(G7,m,g)l = 4,417×C (m−11) 16 For m≥18, g=18, is N(G7,m,g)l = 2,835×C (m−12) 17 For m≥19, g=19, is N(G7,m,g)l = 210×C (m−13) 18 For m≥20, g=20, is N(G7,m,g)l = 21×C (m−14) 19 For m≥21, g=21, is N(G7,m,g)l = C (m−15) 20 Note that themultiplierforg=6is thesameas themultiplier of t= 6 in Ansori et al. (2021), as well as g=7 with t= 7, and so on until g=21 with t=21. However, the formulas are dierent because in Ansori et al. (2021) the formula are for connected vertex labeled graph without loops while in this study is for connected vertices labeled graph without parallel edges. For example, for g=8, N(G7,m,g)l = 30,765×C (m−2) 7 , while in Ansori et al. (2021), for t= 8, N(G7,m,8) = 30,765×C (m−1) 7 . 4. CONCLUSIONS Based on the above reasoning, we may conclude that the num- ber of vertices in a labeled connected graph of order seven with no parallel edges is N(G7,m,g)l= 6,727×Cm6 for g= 6, while for 7≤g≤ 21, N(G7,m,g)l= kg C (m−(g−6)) g−1 , where k7 =30,160, k8 = 30,765, k9 =21,000, k10 =28,364, k11= 26,880, k12 =26,460 , k13 = 20,790, k14 =10,290, k15 = 8,022, k16 = 2,940, k17 =4,417, k18 = 2,835, k19 = 210, k20 = 21, k21 = 1. 5. ACKNOWLEDGMENT The Research Center Universitas Lampung funded this study under Postgraduate research grant project No. 815/UN26. 21/PN/2022, and the authors gratefully acknowledge the fun- ding. REFERENCES Agnarsson, G.andR.Greenlaw(2006). GraphTheory: Modeling, Applications, and Algorithms. Prentice-Hall, Inc. Al Etaiwi, W. M. (2014). Encryption Algorithm Using Graph Theory. JournalofScienticResearchandReports,3(19); 2519– 2527 Álvarez, M. C. and D. Ehnts (2015). 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Journal of Physics: Conference Series, 1338; 012043 © 2022 The Authors. Page 399 of 399 INTRODUCTION OBSERVATION AND INVESTIGATION RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGMENT