A simple graph $G$ with edge-connectivity $\lambda(G)$ and minimum degree $\delta(G)$ is maximally edge connected if $\lambda(G)=\delta(G)$. In 1964, given a non-increasing sequence $\pi=(d_{1},\ldots,d_{n})$, Jack Edmonds showed that there realization of $\pi$ $k$-edge-connected only $d_{n}\geq k$ $\sum_{i=1}^{n}d_{i}\geq 2(n-1)$ when $d_{n}=1$. We strengthen Edmonds's result by showing $G_{0}...

In this paper, we show that every ( 3 k − ) -edge-connected graph G , under a certain degree condition, can be edge-decomposed into factors 1 … such for each vertex v ∈ V i | d / < where ≤ . As an application, deduce 6 three 2 and with unless has exactly one z ⁄ ≡ 0 Next, odd- have the same parity is odd positive integer Finally, give sufficient edge-connectivity condition to factor F specified...

A multigraph is exactly k-edge-connected if there are exactly k edgedisjoint paths between any pair of vertices. We characterize the class of exactly 3-edge-connected graphs, giving a synthesis involving two operations by which every exactly 3-edge-connected multigraph can be generated. Slightly modi ed syntheses give the planar exactly 3-edge-connected graphs and the exactly 3-edge-connected g...

Let G be a connected graph with maximum degree ∆(G). The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. We say that G is maximally irregular if t(G) = ∆(G). The purpose of this note, apart from pointing out that every highly irregular graph is maximally irregular, is to establish upper bounds on the size of maximally irregular graphs and max...

Let G = (V,E) be a multigraph (it has multiple edges, but no loops). We call G maximally edge-connected if λ(G) = δ(G), and G super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edgeconnectivity λ′(G) of G is the minimum number of edges whose removal disconnects G into non-trivial components. If λ′(G) achieves the upper bound of restricted ...

Let G be a (multi)graph of order n and let u,v vertices G. The maximum number internally disjoint u–v paths in is denoted by κG(u,v), the edge-disjoint λG(u,v). average connectivity defined κ¯(G)=∑κG(u,v)∕n2, edge-connectivity λ¯(G)=∑λG(u,v)∕n2, where both sums run over all unordered pairs {u,v}⊆V(G). A graph called ideally connected if κG(u,v)=min{deg(u),deg(v)} for {u,v} We prove that every m...

We focus on the algorithm underlying the main result of [6]. This is an algebraic formula to generate all connected graphs in a recursive and efficient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequentl...