A graph will be assumed to be finite and unoriented, with no loops or multiple edges; if multiple edges are to be allowed, the term multigraph will be used. A graph or multigraph. will be called k-connected if at least k vertices and their incident edges must be removed to disconnect it (a complete graph is considered to be k-connected for any k). A block (respectively, multiblock) is a 2-conne...

In this paper, we prove that in a multigraph whose density Γ exceeds the maximum vertex degree ∆, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with Γ > ∆ + 1, the size of any cluster is bounded from the above by a function, which depends on ∆ and Γ only. Finally, we show that two well-known lower bounds for the chromatic...

Abstract Multivariate networks comprising several compositional and structural variables can be represented as multigraphs by various forms of aggregations based on vertex attributes. We propose a framework to perform exploratory confirmatory multiplexity analysis aggregated in order find relevant associations between edge The exploration is performed comparing frequencies the different edges w...

Eulerian Extension (EE) is the problem to make an arcweighted directed multigraph Eulerian by adding arcs of minimum total cost. EE is NP-hard and has been shown fixed-parameter tractable with respect to the number of arc additions. Complementing this result, on the way to answering an open question, we show that EE is fixedparameter tractable with respect to the combined parameter “number of c...

We tackle the problem of graph transformation with a particular focus on node cloning. We propose a graph rewriting framework where nodes can be cloned zero, one or more times. A node can be cloned together with all its incident edges, with only the outgoing edges, with only the incoming edges or without any of the incident edges. We thus subsume previous works such as the sesqui-pushout, the h...

Unitary coined discrete-time quantum walks (UCDTQW) constitute a universal model of computation, meaning that any computation done by general purpose computer can either be using the UCDTQW framework. In last decades, great progress has been made in this field developing walk-based algorithms outperform classical ones. However, current computers work based on circuit and mapping from one to oth...

We consider a preferential attachment process in which multigraph is built one node at time. The number of edges added stage t, emanating from the new node, given by some prescribed function f(t), generalising model considered Kleinberg and 2005 where f was presumed constant. show that if f(t) asymptotically bounded above below linear functions then with probability 1 infinite limit will be iso...

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), m, n ≥ k2 , m + n > k, and k(p + q) = 2mn for k ≡ 0(mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) = λmn, m, n ≥ k, (resp., m,n ≥ 3k/2) for k ≡ 0(mod 4) (re...