Showing papers on "Path graph published in 1971"
TL;DR: The main purpose of as mentioned in this paper is to prove that every n -connected graph with a 1-factor has at least as many as n (n − 2)( n − 4) … 4 · 2, (or: n ( n − 2/(n − 4)/n − 5 · 3) 1-factors.
18 citations
TL;DR: Systematic generation of a specific class of permutations fundamental to scheduling problems is described, and it is very easy to derive all possible rosary permutations from the 1/2(n
Abstract: Systematic generation of a specific class of permutations fundamental to scheduling problems is described.In a nonoriented complete graph with n vertices, Hamiltonian circuits equivalent to 1/2(n - 1)! specific permutations of n elements, termed rosary permutations, can be defined. Each of them corresponds to two circular permutations which mirror-image each other, and is generated successively by a number system covering 3·4· ··· ·(n - 1) sets of edges. Every set of edges {ek}, 1 ≤ ek ≤ k, 3 ≤ k ≤ n - 1 is determined recursively by constructing a Hamiltonian circuit with k vertices from a Hamiltonian circuit with k - 1 vertices, starting with the Hamiltonian circuit of 3 vertices. The basic operation consists of transposition of a pair of adjacent vertices where the position of the pair in the permutation is determined by {ek}. Two algorithms treating the same example for five vertices are presented.It is very easy to derive all possible n! permutations from the 1/2(n - 1)! rosary permutations by cycling the permutations and by taking them in the reverse order—procedures which can be performed fairly efficiently by computer.
10 citations
TL;DR: In this paper, it was shown that if m ≥ 3, there exists exactly one connected graph having m + 2 vertices and automorphism group S m, the symmetric group of degree m.
3 citations
TL;DR: In this paper, the problem of determining a point on a tree having the property that the sum of the products of the intensities of its vertices by the corresponding distances to that point is a minimum is solved.
Abstract: The following problem is solved: determine a point on a tree having the property that the sum of the products of the intensities of its vertices by the corresponding distances to that point is a minimum. The proposed algorithm is reduced to the stepwise application to the tree of truncation of its vertices. A feasible interpretation of the problem is given.