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

Enabling cooperation, resource allocation and receiver selection across cells: Complementary fractional frequency reuse

IEEE 24th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications, PIMRC 2013; London; United Kingdom; 8 September 2013 through 11 September 2013For a multi-cell multiple access channel, we develop a comprehensive cooperative communication framework: we propose a novel complementary fractional frequency reuse (FFR) strategy tailored specifically for pairwise user cooperation, also taking into account cell sectoring. This strategy allows the cell edge users not only to pool their resources and cooperate across cells, but also to choose the best receiver. We divide the users into cooperating inner and outer user pairs, and assign each pair orthogonal resources using OFDMA. We employ pairwise bidirectional cooperation based on block Markov superposition encoding among user pairs. We derive the achievable rates, while taking into account the geometry dependent interference at the users and the receiver. We find the jointly optimal power allocation, partner selection and receiver selection strategies that maximize the sum rate of the system. We then propose a heuristic matching algorithm, which operates based only on user and receiver locations. We compare the performance of our proposed strategies with several non-cooperative models, and demonstrate that the sum rate can nearly be doubled, while using the same resources. © 2013 IEEE