On the number of edges in cycletrees

Mathematical and Computer Modelling, 1991

It is shown that the number of labeled spanning trees of a connected graph is equal to the determinant of the intersection matrix for an integral basis of its cycles. The one-skeletons of a convex polyhedron and its dual have the same number of spanning trees. An integral cycle basis is constructed for graphs possessing rotational symmetry. Certain sequences of a priori algebraic numbers are seen to be computable using integers only.

Journal of Combinatorial Theory, Series B, 1988

Discrete Applied Mathematics, 2020

Hamiltonicity of graphs possessing symmetry has been a popular subject of research, with focus on vertex-transitive graphs, and in particular on Cayley graphs. In this paper, we consider the Hamiltonicity of another class of graphs with symmetry, namely covering graphs of trees. In particular, we study the problem for covering graphs of trees, where the tree is a voltage graph over a cyclic group. Batagelj and Pisanski were first to obtain such a result, in the special case when the voltage assignment is trivial; in that case, the covering graph is simply a Cartesian product of the tree and a cycle. We consider more complex voltage assignments, and extend the results of Batagelj and Pisanski in two different ways; in these cases the covering graphs cannot be expressed as products. We also provide a linear time algorithm to test whether a given assignment satisfies these conditions.

Open Journal of Discrete Mathematics

In this paper, we obtain explicit formulae for the number of 7-cycles and the total number of cycles of lengths 6 and 7 which contain a specific vertex v i in a simple graph G, in terms of the adjacency matrix and with the help of combinatorics.

SIAM Journal on Discrete Mathematics, 2018

Let L be a set of positive integers. We call a (directed) graph G an L-cycle graph if all cycle lengths in G belong to L. Let c(L, n) be the maximum number of cycles possible in an n-vertex L-cycle graph (we use c(L, n) for the number of cycles in directed graphs). In the undirected case we show that for any fixed set L, we have c(L, n) = Θ(n k/ ) where k is the largest element of L and 2 is the smallest even element of L (if L contains only odd elements, then c(L, n) = Θ(n) holds.) We also give a characterization of L-cycle graphs when L is a single element. In the directed case we prove that for any fixed set L we have c(L, n) = (1 + o( ))( n-1 k-1 ) k-1 , where k is the largest element of L. We determine the exact value of c({k}, n) for every k and characterize all graphs attaining this maximum.

There exist many algorithms for producing the spanning trees of a graph with better time and space complexities. In this research study, we are presenting a study on number of spanning trees and a technique based on the basic cycle to find the number of spanning trees and also the structure of all the spanning trees of a labeled and undirected graph.

2009

Let t(G) denote the number of spanning trees of a graph G. A chain of two connected vertices u,v(dG(u),dG(v) � 3) in G, denoted by Lk, is defined as a path of G and dG(p) = 2 for all p 2 V (Lk) { u,v}, where k is the length of the path. In this paper, we investigate the relationship between t(G) and Lk of a graph G. In particular, the relationship between t(G) and Lk of �-optimal graph G is considered.

2000

The problem studied in this paper is that of ÿnding the maximum number of Hamiltonian cycles in a graph with a given number of vertices and edges. The main results are a lower bound and an upper bound, both given by closedform formulas, for the maximum number of Hamiltonian cycles in a graph with a given number of vertices and edges.

Discrete Applied Mathematics, 1997

We prove that if a graph G on n > 32 vertices is hamiltonian and has two nonadjacent vertices u and u with d(u) + d(u) 3 n + z where z = 0 if n is odd and z = 1 if n is even, then G contains all cycles of length m where 3 < m < 1/5(n + 13).

Information Processing Letters, 2005

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ore's theorem directly follows from our result.