The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm can be considered as a greedy version of Bm, whereas Em and Fm are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm Am in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm Am) to 0.08% over optimum (algorithm Em). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm Am) to 0.45% over optimum (algorithm Fm). Better quality of solutions demands higher computation times, thus the new algorithms provide a good compromise between quality of solutions and computation time.
Summary
Quartiles are usually introduced early, often in primary school together with box‐and‐whisker plots. Various methods are used, and in lack of explanation in many textbooks at elementary level, this leads to unnecessary confusion. We discuss some elementary methods that are consistent with the most common definition and are also easy to understand.
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