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Yayın Four-cycled graphs with topological applications(Birkhauser Verlag AG, 2012-03) Bıyıkoğlu, Türker; Civan, YusufWe call a simple graph G a 4-cycled graph if either it has no edges or every edge of it is contained in an induced 4-cycle of G. Our interest on 4-cycled graphs is motivated by the fact that their clique complexes play an important role in the simple-homotopy theory of simplicial complexes. We prove that the minimal simple models within the category of flag simplicial complexes are exactly the clique complexes of some 4-cycled graphs. We further provide structural properties of 4-cycled graphs and describe constructions yielding such graphs. We characterize 4-cycled cographs, and 4-cycled graphs arising from finite chessboards. We introduce a family of inductively constructed graphs, the external extensions, related to an arbitrary graph, and determine the homotopy type of the independence complexes of external extensions of some graphs.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.
