This paper addresses two puzzles in the domain of verb cluster formation and proposes a solution in terms of rule ordering. The first puzzle is the socalled extraposition paradox where extraposition can target a VP that is part of a verb cluster only if the VP is topicalized but not when the VP remains clause-final. I propose that verb cluster formation takes place at PF under adjacency and thu...

This paper surveys the recent progress on the graph algorithms for solving network connectivity problems such as the extreme set problem, the cactus representation problem, the edge-connectivity augmentation problem and the source location problem. In particular, we show that efficient algorithms for these problems can be designed based on maximum adjacency orderings.

Given a connected irregular graph G of order n, write μ for the largest eigenvalue of its adjacency matrix, ∆ for its maximum degree, andD for its diameter. We prove that ∆− μ > 1 (D + 2)n and this bound is tight up to a constant factor. This improves previous results of Stevanović and Zhang, and extends a result of Alon and Sudakov.

Let G be a simple graph of order n, let λ1(G), λ2(G), . . . , λn(G) be the eigenvalues of the adjacency matrix of G. The Esrada index of G is defined as EE(G) = ∑n i=1 e i. In this paper we determine the unique graph with maximum Estrada index among bicyclic graphs with fixed order.

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.

NMR resonance peak assignment is one of the key steps in solving an NMR protein structure. The assignment process links resonance peaks to individual residues of the target protein sequence, providing the prerequisite for establishing intra- and inter-residue spatial relationships between atoms. The assignment process is tedious and time-consuming, which could take many weeks. Though there exis...

Let ∆(G), ∆ for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree ∆, where ∆ ≥ n+1 2 (resp., ∆ ≥ n 2 +1 and ∆ ≥ n+3 2 ) are determ...

Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conject...

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph H = (V, E) with n= |V |, m= |E| and p = ∑ e∈E |e| the fastest known algorithm to compute a global minimum cut in H runs in O(np) time for the uncapacitated case, and in O(np+ n2 log n) time for the capacitated case. We show the following new results. • Given an uncapacitated hypergraph H and an intege...

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L2(G), the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. ...