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Yayın Algebraic connectivity and degree sequences of trees(Elsevier Science Inc, 2009-01-15) Bıyıkoğlu, Türker; Leydold, JosefWe investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.Yayın Graphs with given degree sequence and maximal spectral radius(Electronic Journal of Combinatorics, 2008-09-15) Bıyıkoğlu, Türker; Leydold, JosefWe describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization.Yayın Some notes on spectra of cographs(Charles Babbage Res Ctr, 2011-07) Bıyıkoğlu, Türker; Simic, Slobodan K.; Stanic, ZoranA cograph is a P-4-free graph. We first give a short proof of the fact that 0 (-1) belongs to the spectrum of a connected cograph (with at least two vertices) if and only if it contains duplicate (resp. coduplicate) vertices. As a consequence, we next prove that the polynomial reconstruction of graphs whose vertex-deleted subgraphs have the second largest eigenvalue not exceeding root 5-1/2 is unique.Yayın Semiregular trees with minimal Laplacian spectral radius(Elsevier Inc, 2010-04-15) Bıyıkoğlu, Türker; Leydold, JosefA semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence.
