Leaf-removal process has been widely researched and applied in many mathematical and physical fields to help understand the complex systems. A lot of problems including the minimum vertex-cover are deeply related to t...
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Leaf-removal process has been widely researched and applied in many mathematical and physical fields to help understand the complex systems. A lot of problems including the minimum vertex-cover are deeply related to this process and the leaf-removal cores. In this paper, based on the structural features of the leaf-removal cores, a method named core influence is proposed to break the graphs into no-leaf-removal-core ones, which takes advantages of identifying some significant nodes by localized and greedy strategy. By decomposing the minimum vertex-cover problem into the leaf-removal core breaking process and maximal matching of the remaining graphs, it is proved that any minimum vertex-cover of the whole graph can be located into these two processes and the best boundary is achieved at the transition point. Compared with other node importance indices, the core influence method could break down the leaf-removal cores much faster and get the no-core graphs by removing fewer nodes from the graphs. Also, the vertex-cover numbers resulted from this method are lower than those of existing node importance measurements, and compared with the exact minimum vertex-cover numbers, this method performs appropriate accuracy and stability at different scales. This research provides a new localized greedy strategy to break the hard leaf-removal cores efficiently and heuristic methods could be constructed to promote the comprehension of the intrinsic hardness of NP problems.
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The c...
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The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.
A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disor...
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A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs. It is illustrated with two combinatorial optimization problems, the vertex-cover and colouring problems, for which the presence of replica symmetry breaking phases is taken into account. Applications include the analysis of models on adaptive graph structures.
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