Let G = (V,E) be a k-edge-connected graph with edge-costs {c(e) : e ∈ E} and minimum degree d. We show by a simple and short proof, that for any integer l with d k ≤ l ≤ d (

We present new edge splitting-off results maintaining all-pairs edge-connectivities of an undirected graph. We first give an alternate proof of Mader’s theorem, and use it to obtain a deterministic Õ(m + rmax · n2)-time complete edge splitting-off algorithm for unweighted graphs, where rmax denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow b...

Let G be a connected graph with vertex set V (G), order n = |V (G)|, minimum degree δ and edge-connectivity λ. Define the inverse degree of G as R(G) = ∑ v∈V (G) 1 d(v) , where d(v) denotes the degree of the vertex v. We show that if R(G) < 2 + 2 δ(δ + 1) + n− 2δ (n− δ − 2)(n− δ − 1) , then λ = δ. We also give an analogous result for triangle-free graphs.

In this paper, we introduce generalized connectivity in L-fuzzy topological spaces by à Lukasiewicz logic and prove K. Fan’s theorem.

The strong product G1 £G2 of graphs G1 and G2 is the graph with V (G1) × V (G2) as the vertex set, and two distinct vertices (x1, x2) and (y1, y2) are adjacent whenever for each i ∈ {1, 2} either xi = yi or xiyi ∈ E(Gi). In this note we show that for two connected graphs G1 and G2 the edge-connectivity λ(G1£G2) equals min{δ(G1£ G2), λ(G1)(|V (G2)|+2|E(G2)|), λ(G2)(|V (G1)|+2|E(G1)|)}. In additi...

We show that for all k ≤ −1 an interval graph is −(k + 1)Hamilton-connected if and only if its scattering number is at most k. We also give an O(n +m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the O(n) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is...

let $g$ be a simple connected graph. the edge-wiener index $w_e(g)$ is the sum of all distances between edges in $g$, whereas the hyper edge-wiener index $ww_e(g)$ is defined as {footnotesize $w{w_e}(g) = {frac{1}{2}}{w_e}(g) + {frac{1}{2}} {w_e^{2}}(g)$}, where {footnotesize $ {w_e^{2}}(g)=sumlimits_{left{ {f,g} right}subseteq e(g)} {d_e^2(f,g)}$}. in this paper, we present explicit formula fo...