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**1 - 3**of**3**### How to Construct the Characteristic Polynomial of a Tree's Adjacency Matrix

, 1996

"... We describe a simple, O(n 2 log(n)) algorithm to find the characteristic polynomial of the adjacency matrix of any tree. An example is given showing the algorithm applied to a 13-vertex tree. key words: tree, adjacency matrix, characteristic polynomial. 1 Introduction Let G = (V; E) be an undire ..."

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We describe a simple, O(n 2 log(n)) algorithm to find the characteristic polynomial of the adjacency matrix of any tree. An example is given showing the algorithm applied to a 13-vertex tree. key words: tree, adjacency matrix, characteristic polynomial. 1 Introduction Let G = (V; E) be an undirected graph with vertices V = (v 1 ; : : : ; v n ) and edge set E. Edges occur only between pairs of distinct vertices, and between any pair of vertices there is at most one edge. The adjacency matrix A = [a ij ] of G is the n \Theta n 0-1 matrix for which a ij = 1 if and only if v i is adjacent to v j (that is, there is an edge between v i and v j ). In this paper, a graph is always a tree (i.e., a connected, acyclic graph) or a forest (i.e., a disjoint collection of trees). The first author thanks CNPq and Clemson University for the support to visit UFRGS while this research was performed. y The second author also thanks CNPq for support during this period. D. P. Jacobs and V. Trev...

### COMPLEXITY THEORETIC ASPECTS OF PLANAR RESTRICTIONS AND OBLIVIOUSNESS

, 2006

"... In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomial-sized ci ..."

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In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomial-sized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the well-known circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NL-complete). We prove that this problem logspace-reduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining