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

Abstract

Given 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 graphs obtained may be simple or not simple, connected or disconnected. A graph G(V,E) is called simple if G(V,E) not containing loops nor paralel edges. An edge which has the same end vertex is called a loop, and paralel edges are two or more edges which connect the 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.

Enumerate the Number of Vertices Labeled Connected Graph of Order Seven Containing No Parallel Edges

“…However, the formulas are di erent 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(G 7,m, g ) l = 30,765×C (m−2) 7 , while in Ansori et al (2021) , for t= 8, N(G 7,m,8 ) = 30,765×C (m−1) 7 .…”

“…The following results are obtained by doing the similar manner: Note that the multiplier for g= 6 is the same as the multiplier 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 di erent 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(G 7,m, g ) l = 30,765×C (m−2) 7 , while in Ansori et al (2021) , for t= 8, N(G 7,m,8 ) = 30,765×C (m−1) 7 .…”

“…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.…”