arXiv subjects

Türker Bıyıkoğlu

Publications and source records attributed to Türker Bıyıkoğlu.

4 recordsLinked to original sources

arXiv2025

Graph entropy, degree assortativity, and hierarchical structures in networks

We connect several notions relating the structural and dynamical properties of a graph. Among them are the topological entropy coming from the vertex shift, which is related to the spectral radius of the graph's adjacency matrix, the Randić index, and the degree assortativity. We show that, among all connected graphs with the same degree sequence, the graph having maximum entropy is characterized by a hierarchical structure; namely, it satisfies a breadth-first search ordering with decreasing degrees (BFD-ordering for short). Consequently, the maximum-entropy graph necessarily has high degree assortativity; furthermore, for such a graph the degree centrality and eigenvector centrality coincide. Moreover, the notion of assortativity is related to the general Randić index. We prove that the graph that maximizes the Randić index satisfies a BFD-ordering. For trees, the converse holds as well. We also define a normalized Randić function and show that its maximum value equals the difference of Shannon entropies of two probability distributions defined on the edges and vertices of the graph based on degree correlations.

math.CO↗

arXiv2023

Bounding the collapsibility number of simplicial complexes and graphs

We introduce and study a new combinatorial invariant the theta-number $θ(X)$ of simplicial complexes, and prove that the inequality $\mathcal{C}(X)\leq θ(X)$ holds for every simplicial complex $X$, where $\mathcal{C}(X)$ denotes the collapsibility number of $X$. We display the advantages of working with the theta-number. Its purely combinatorial formulation enables us to verify the validity of the existing bounds on both Leray and collapsibility numbers as well as provide new bounds involving other parameters. We show that the theta-number, collapsibility and Leray numbers of a vertex decomposable simplicial complex are all equal. Moreover, we prove that the theta-number of the independence complex of a graph $G$ is closely related to its induced matching number $im(G)$ as it happens to the Leray number of such complexes. We identify graph classes where they are equal, and otherwise provide upper bounds involving it. In particular, we prove that the theta-number is bounded from above by $2\sqrt{n\cdot im(G)}$ for every $n$-vertex graph $G$, and in the case of $2K_2$-free graphs, we lower this bound to $2\log n$. Furthermore, we verify that the theta-number is contraction minor monotone on the underlying graph.

math.CO↗

arXiv2019

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.

cs.DS↗

arXiv2016

Projective dimension of (hyper)graphs and the Castelnuovo-Mumford regularity of bipartite graphs

We prove that the projective dimension of any (hyper)graph can be bounded from above by the (Castelnuovo-Mumford) regularity of its Levi graph (or incidence bipartite graph). This in particular brings the use of regularity's upper bounds on the calculation of projective dimension of (hyper)graphs. When G is just a (simple) graph, we prove that there exists an induced subgraph H of G such that prod-dim(G)=reg(S(H)), where S(H) is the subdivision graph of H. Moreover, we show that known upper bounds on prod-dim(G) involving domination parameters are in fact upper bounds to reg(S(G)).

math.CO↗