A Textbook of Graph Theory

Modelling Dominance Hierarchies Under Winner and Loser Effects

“…In particular, each individual scores increasingly better against those that are lower in rank, and increasingly worse against those individuals that are higher in rank. Figure 2,a) shows the values of the index of linearity K calculated by equation (6). We observe that K is close to 1 for all parameter combinations considered, indicating a near linear hierarchy in almost all of the cases.…”

“…Below we show that the original definition of d as the number of circular triads, in the case where there was a single contest between each pair of individuals, is a special case of our definition from (6). For the single contest case, d is the total number of triples minus the total number of transitive triples [6].…”

“…For the single contest case, d is the total number of triples minus the total number of transitive triples [6]. The number of transitive triples is…”

Operations on Digraphs and Digraph Folding

“…Let f Ð( )be defined by f{u 4 }={u 2 }, f{e 1 ,e 2 } = {e 3 ,e 4 } and g Ð( ) be defined by g{v 1 } = { v 3 } , g{e 6 ,e 8 }= {e 5 ,e 7 }. Then the cartesian product dimap h = f x g : D 1 x D 2 D 1 x D 2 is defined as follows: h{(u 4 ,v 2 ),(u 3 ,v 1 )}={(u 2 ,v 2 ),(u 3 ,v 3 )}, and so on.…”

“…Then the cartesian product dimap h = f x g : D 1 x D 2 D 1 x D 2 is defined as follows: h{(u 4 ,v 2 ),(u 3 ,v 1 )}={(u 2 ,v 2 ),(u 3 ,v 3 )}, and so on. Also, h{((u4,v2),(u1,v2)), ((u3,v1),(u3,v2))}= {((u2,v2),(u1,v2)), ((u3,v3),(u3,v2))}, and so on, see Figure 6.…”