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Ring-current maps give a direct pictorial representation of molecular aromaticity. They can be computed at levels ranging from empirical to full ab initio and DFT. For benzenoid hydrocarbons, Hückel–London (HL) theory gives a remarkably good qualitative picture of overall current patterns, and a useful basis for their interpretation. This paper describes an implemention of Aihara’s algorithm for computing HL currents for a benzenoid (for example) by partitioning total current into its constituent cycle currents. The Aihara approach can be used as an alternative way of calculating Hückel–London current maps, but more significantly as a tool for analysing other empirical models of induced current based on conjugated circuits. We outline an application where examination of cycle contributions to HL total current led to a simple graph-theoretical approach for cycle currents, which gives a better approximation to the HL currents for Kekulean benzenoids than any of the existing conjugated-circuit models, and unlike these models it also gives predictions of the HL currents in non-Kekulean benzenoids that are of similar quality.

We establish exact values for the$2$ -limited broadcast domination number of various grid graphs, in particular$C_m\\square C_n$for$3 \\leq m \\leq 6$and all$n\\geq m$ ,$P_m \\square C_3$for all$m \\geq 3$ , and$P_m \\square C_n$for$4\\leq m \\leq 5$and all$n \\geq m$ . We also produce periodically optimal values for$P_m \\square C_4$and$P_m \\square C_6$for$m \\geq 3$ ,$P_4 \\square P_n$for$n \\geq 4$ , and$P_5 \\square P_n$for$n \\geq 5$ . Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

We establish exact values for the2 -limited broadcast domination number of various grid graphs, in particularC_(m)□ C_(n)for3 ≤ m ≤ 6and alln≥ m ,P_(m) □ C₃for allm ≥ 3 , andP_(m) □ C_(n)for4≤ m ≤ 5and alln ≥ m . We also produce periodically optimal values forP_(m) □ C₄andP_(m) □ C₆form ≥ 3 ,P₄ □ P_(n)forn ≥ 4 , andP₅ □ P_(n)forn ≥ 5 . Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

We establish exact values for the 2-limited broadcast domination number of various grid graphs, in particular Cm□Cn for 3 ≤ m ≤ 6 and all n ≥ m, Pm□C3 for all m ≥ 3, and Pm□Cn for 4 ≤ m ≤ 5 and all n ≥ m. We also produce periodically optimal values for Pm□C4 and Pm□C6 for m ≥ 3, P4□Pn for n ≥ 4, and P5□Pn for n ≥ 5. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

We establish upper and lower bounds for the 2-limited broadcast domination number of various grid graphs, in particular the Cartesian product of two paths, a path and a cycle, and two cycles. The upper bounds are derived by explicit constructions. The lower bounds are obtained via linear programming duality by finding lower bounds for the fractional 2-limited multipacking numbers of these graphs.

We establish exact values for the \\(2\\)-limited broadcast domination number of various grid graphs, in particular \\(C_m\\square C_n\\) for \\(3 \\leq m \\leq 6\\) and all \\(n\\geq m\\), \\(P_m \\square C_3\\) for all \\(m \\geq 3\\), and \\(P_m \\square C_n\\) for \\(4\\leq m \\leq 5\\) and all \\(n \\geq m\\). We also produce periodically optimal values for \\(P_m \\square C_4\\) and \\(P_m \\square C_6\\) for \\(m \\geq 3\\), \\(P_4 \\square P_n\\) for \\(n \\geq 4\\), and \\(P_5 \\square P_n\\) for \\(n \\geq 5\\). Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degree, except when simple arithmetic forces the edge-chromatic number to be one greater than the maximum degree. For Mycielski graphs, we strengthen an old result that the graphs are Hamiltonian by showing that they are Hamilton-connected (except M(3), which is a cycle). For Keller graphs G(d), we establish, in all cases, the exact value of the chromatic number, the edge-chromatic number, and the independence number, and we get the clique covering number in all cases except 5 <= d <= 7. We also investigate Hamiltonian decompositions of Keller graphs, obtaining them up to G(6).

The edge-reconstruction number ern\\((G)\\) of a graph \\(G\\) is equal to the minimum number of edge-deleted subgraphs \\(G-e\\) of \\(G\\) which are sufficient to determine \\(G\\) up to isomorphsim. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.

A polytope is called regular-faced if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by G. Blind and R. Blind BlBl2,roswitha,roswitha2. The last class of such polytopes is the one which consists of polytopes obtained by removing a set of non-adjacent vertices (an independent set) of the 600-cell. These independent sets are enumerated up to isomorphism and it is determined that the number of polytopes in this last class is \\(314,248,344\\).

A d-dimensional Keller graph has vertices which are numbered with each of the 4^sup d^ possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a \"high speed computer the size of a major galaxy\". This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124. [PUBLICATION ABSTRACT]