Centroid

Centroid --- центроид.

A branch of a tree [math]\displaystyle{ T }[/math] at a vertex [math]\displaystyle{ v }[/math] is a maximal subtree [math]\displaystyle{ T_{v} }[/math] of [math]\displaystyle{ T }[/math], in which the degree of [math]\displaystyle{ v }[/math] is unity. Therefore, the number of branches at [math]\displaystyle{ v }[/math] is [math]\displaystyle{ deg(v) }[/math]. The branch-weight centroid number of a vertex [math]\displaystyle{ v }[/math] in a tree [math]\displaystyle{ T }[/math], denoted by [math]\displaystyle{ bw(v) }[/math] is the the maximum size of any branch at [math]\displaystyle{ v }[/math]. A vertex [math]\displaystyle{ v }[/math] of a tree [math]\displaystyle{ T }[/math] is a centroid vertex of [math]\displaystyle{ T }[/math] if [math]\displaystyle{ v }[/math] has minimum branch-weight centroid number. The centroid of [math]\displaystyle{ T }[/math] consists of its set of centroid vertices. Jordan (1869) has proved the following theorem.

Theorem. The centroid of a tree consists of either a single vertex or a pair of adjacent vertices.

See also

• Slater number.