On the Edge Balance Index of Flux Capacitor, Centipede, L-Product Graphs

1. On the Edge Balance Index of Flux Capacitor, Centipede, L-Product Graphs Meg Galiardi and Dan Perry Abstract- Let G be a graph with vertex set V (G) and edge set E(G), and let Z2 = {0, 1}. A labeling f : E(G) → Z2 of a graph G is said to be edge-friendly if {|ef(0) − ef (1)| ≤ 1}. An edge-friendly labeling f induces a partial vertex labeling f+ : V (G) → Z2 defined by f+(x) = 0 if the number of edges labeled by 0 incident on x is more than the number of edges labeled by 1 incident on x. Similarly, f+(x) = 1 if the number of edges labeled by 1 incident on x is more than the number of edges labeled by 0 incident on x. f+(x) is not define if the number of edges labeled by 1 incident on x is equal to the number of edges labeled by 0 incident on x. For iЄZ2, let vf(i) = card{vЄV (G) : f+(v) = i} and ef(i) = card{eЄE(G) : f(e) = i}. The edge-balance index set of the graph G, EBI(G), is defined as {|vf(0) − vf(1)| : the edge labeling f is edge-friendly}. Graph edges are labeled by either 0 or 1. For an edge-friendly labeling, the total difference in 0 and 1 edges can be at most 1. Vertices are then labeled 0 if the number of 0-edges adjacent to it is more than the number of 1-edges. Vertices are labeled 1 if the number of 1-edges adjacent to it is more than the number of 0-edges. If there are an equal number of 0 and 1 edges adjacent to a vertex, it remains unlabeled. The edge balance index (EBI) is the set of all possible difference in 1 and 0 vertices such that the graph has an edge-friendly labeling. A flux capacitor graph is composed of two different types of graphs, a star graph and a cycle. A star, St(n), consists of a center vertex, s0, and n other vertices each connected to the center. A cycle, Cm, consists of m vertices each connected to 2 others to form a cycle where m≥ 3. A flux capacitor graph, FC(n,m), is a St(n) graph where on each outer vertex there is a Cm graph. An L-product graph of star by cycle graph is the same as a flux capacitor graph, the only difference being there is an additional cycle, Cm on the center vertex of the star. It is represented as St(n)XLCm. A Centipede Graph is composed of two cycle graphs. A cycle, Cn, consists of n vertices each connected to 2 others to form a cycle where n≥3. There is then a cycle, Cm, attached to each vertex of Cn. Ce(n,m) is the abbreviation. EBI(FC(n,m)) = {0,1,. . .,n-1} if m is odd, {0,1,. . .,n-1} if n is even and m is even, {0,1,. . .,n} if n is odd and m is even. EBI(St(n)XLCm) = {0, 1...n + 1} when m is odd {0, 1...n} when n is even and m is even {0, 1...n + 1} when n is odd and m is even EBI(Ce(n,3)) = {0,1} if n=3, {0,1,. . .,k} if n=2k is even, kЄZ {0,1,. . .,k-1} if n=2k+1 is odd, kЄZ, k≥2. If n=2k+1 is odd, where kЄZ, EBI(Ce(n,4)) = {0,1,. . ., 3 + 8[(k-1)/5]} if k-1=0 in Mod 5, {0,1,. . ., 4 + 8[(k-2)/5]} if k-1=1 in Mod 5. {0,1,. . ., 6 + 8[(k-3)/5]} if k-1=2 in Mod 5. {0,1,. . ., 8[(k+1)/5]} if k-1=3 in Mod 5. {0,1,. . ., 9 + 8[(k-5)/5]} if k-1=4 in Mod 5. If n=2k is even, where kЄZ, EBI(Ce(n,4)) = {0,1,. . ., 3 + 8[(k-2)/5]} if k-2=0 in Mod 5, {0,1,. . ., 4 + 8[(k-3)/5]} if k-2=1 in Mod 5. {0,1,. . ., 6 + 8[(k-4)/5]} if k-2=2 in Mod 5. {0,1,. . ., 8[(k)/5]} if k-2=3 in Mod 5. {0,1,. . ., 9 + 8[(k-6)/5]} if k-2=4 in Mod 5. If m ≥ 5, EBI(Ce(n,m)) = {0,1,. . ., n} if m is odd, {0,1,. . ., n + 1} if m is even and n is odd. {0,1,. . ., n} if m and n are even. Flux Capacitor Graph (FC(3,8)) L-Product of Star by Cycle (St(4)XLC4) Centipede Graph (Ce(7,4))