Related papers: Exponentiation of Graphs
A longstanding problem in spectral graph theory asks for graphs with maximum number of spanning trees among all connected simple graphs with a prescribed number of vertices and edges. Such graphs are called t-optimal graphs. Petingi and…
The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs…
We consider {\em L-graphs}, that is contact graphs of axis-aligned L-shapes in the plane, all with the same rotation. We provide several characterizations of L-graphs, drawing connections to Schnyder realizers and canonical orders of…
Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis…
The exponential family of random graphs represents an important and challenging class of network models. Despite their flexibility, conventionally used exponential random graphs have one shortcoming. They cannot directly model weighted…
We give constructive proofs for the existence of uniquely hamiltonian graphs for various sets of degrees. We give constructions for all sets with minimum 2 (a trivial case added for completeness), all sets with minimum 3 that contain an…
We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all…
We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…
We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we…
Galluccio--Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially…
For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a…
Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…
The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper,…
We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of connected bipartite graphs with given Betti number. By solving these equations…
Exponential-family random graph models (ERGMs) provide a principled and flexible way to model and simulate features common in social networks, such as propensities for homophily, mutuality, and friend-of-a-friend triad closure, through…
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
In this paper, a new concept in graphs namely well-f-coveredness is introduced. We characterize all graphs with such property, whose maximum induced forests are of boundary order. Also we prove several propositions concerning with obtaining…
Graphs are a representation of structured data that captures the relationships between sets of objects. With the ubiquity of available network data, there is increasing industrial and academic need to quickly analyze graphs with billions of…
This paper revisits the study of (minimum) broadcast graphs, i.e., graphs enabling fast information dissemination from every source node to all the other nodes (and having minimum number of edges for this property). This study is performed…
We introduce a notion for hierarchical graph clustering which we call the expander hierarchy and show a fully dynamic algorithm for maintaining such a hierarchy on a graph with $n$ vertices undergoing edge insertions and deletions using…