In this note we give a characterization of the complete bipartite graphs which have an even (odd) [a, b]-factor. For general graphs we prove that an a-edge connected graph G with n vertices and with δ(G) ≥ max{a + 1, an a+b + a − 2} has an even [a, b]-factor, where a and b are even and 2 ≤ a ≤ b. With regard to the edge-connectivity this result is slightly better than one of the similar results...

Let G be a connected graph of order n, minimum degree δ(G), and edge-connectivity κ (G). The graph G ismaximally edge-connected if κ (G) = δ(G) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. A list (d1, . . . , dn) is graphic if there is a graph with vertices v1, . . . , vn such that d(vi) = di for 1 ≤ i ≤ n. A graphic list D is su...

Let G be a finite connected graph. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u + deg v) dG(u, v), where degw is the degree of vertex w and dG(u, v) denotes the distance between u and v in G. In this paper, we give asymptotically sharp upper bounds on the degree distance in terms of order and edge-connectivity.

Let c : E → {1, . . . , k} be an edge colouring of a connected graph G = (V,E). Each vertex v is endowed with a naturally defined pallet under c, understood as the multiset of colours incident with v. If δ(G) > 2, we obviously (for k large enough) may colour the edges of G so that adjacent vertices are distinguished by their pallets of colours. Suppose then that our coloured graph is examined 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.

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, (P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is superif each minimum edge cut is trivial, and it is optimum...

Background: Early diagnosis of patients in the early stages of Alzheimer's, known as mild cognitive impairment, is of great importance in the treatment of this disease. If a patient can be diagnosed at this stage, it is possible to treat or delay Alzheimer's disease. Resting-state functional magnetic resonance imaging (fMRI) is very common in the process of diagnosing Alzheimer's disease. In th...