Dendrimers are the unique chemical trees with maximum spectral radius

It is shown that dendrimers have maximum spectral radius and maximum Collatz-Sinogowitz index among all chemical trees of given size. The result is also generalized for the class of chemical trees with prescribed number of pendant vertices.T. Biyikoglu is supported by Turkish Academy of Sciences through Young Scientist Award Program (TUBA-GEBIP/2009) and by The Scientific and Technological Research Council of Turkey (TUBITAK) TUBITAK Grant 210T173 within the EUROCORES Programme EUROGIGA project GReGAS of the European Science Foundation (ESF). The Austrian participation in GReGAS is not supported by the Austrian Science Fund (FWF)Publisher's Versio

Structure of conflict graphs in constraint alignment problems and algorithms

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.Comment: 22 pages, 6 figure

Geometric representations and symmetries of graphs, maps and other discrete structures and applications in science

Bu proje çalışmasının birinci amacı çizgelerin özdeğer ve özvektör yapılarını çizge özellik ve sabitleri ile ilişkilendirmektir. İkinci amacı biyoloji, bioinformatik, dinamik sistemler, haberleşme, kriptoloji ve sosyal ağlar gibi bir birinden çok farklı alanlardan gelen birbirinden tamamen bağımsız olan temel problemler için çizge kuramı ile özgün modellenmesi ve ortaya çıkan çizge problemlerin çözülmesidir. Üçüncü amacı çizgelerin Castelnuovo-Mumford regülaritesine indirgenmiş eşleşme sayısı ile üstten etkin sınırlar getirmektir.TÜBİTAKPublisher's Versio

Cryptanalysis of Fridrich's chaotic image encryption

We cryptanalyze Fridrich's chaotic image encryption algorithm. We show that the algebraic weaknesses of the algorithm make it vulnerable against chosen-ciphertext attacks. We propose an attack that reveals the secret permutation that is used to shuffle the pixels of a round input. We demonstrate the effectiveness of our attack with examples and simulation results. We also show that our proposed attack can be generalized to other well-known chaotic image encryption algorithms.Scientific and Technological Research Council of Turkey (TUBITAK) [106E143]; Turkish Academy of Sciences [TUBA-GEBIP/2009]Ercan Solak and Cahit Cokal were supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under Project No. 106E143. Turker Biyikoglu was supported by Turkish Academy of Sciences through Young Scientist Award Program (TUBA-GEBIP/2009).Publisher's Versio

Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks

Graphs of Given Order and Size and Minimum Algebraic Connectivity

The structure of connected graphs of given size and order that have minimal algebraic connectivity is investigated. It is shown that they must consist of a chain of cliques. Moreover, an upper bound for the number of maximal cliques of size 2 or larger is derived. (author's abstract)Series: Research Report Series / Department of Statistics and Mathematic

Four-cycled graphs with topological applications

We 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.Both authors are supported by TUBA through Young Scientist Award Program (TUBA-GEBIP/2009-06 and 2008-08)Publisher's Versio

CAMPways: Constrained alignment framework for the comparative analysis of a pair of metabolic pathways

Motivation: Given a pair of metabolic pathways, an alignment of the pathways corresponds to a mapping between similar substructures of the pair. Successful alignments may provide useful applications in phylogenetic tree reconstruction, drug design and overall may enhance our understanding of cellular metabolism.Results: We consider the problem of providing one-to-many alignments of reactions in a pair of metabolic pathways. We first provide a constrained alignment framework applicable to the problem. We show that the constrained alignment problem even in a primitive setting is computationally intractable, which justifies efforts for designing efficient heuristics. We present our Constrained Alignment of Metabolic Pathways (CAMPways) algorithm designed for this purpose. Through extensive experiments involving a large pathway database, we demonstrate that when compared with a state-of-the-art alternative, the CAMPways algorithm provides better alignment results on metabolic networks as far as measures based on same-pathway inclusion and biochemical significance are concerned. The execution speed of our algorithm constitutes yet another important improvement over alternative algorithms. © The Author 2013.TUBITAK (112E137); TÜBA GEBIP 2009 and ESF EUROCORES TUBITAK (210T173