Extremal Trees with Fixed Degree Sequence

A note on extremal trees with degree conditions

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree.

In this paper, we give structural and degree sequence characterizations for a new class of digraphs called directed 3-cycle anchored. A digraph in this class has the property that, for every realization of its degree sequence, there is a directed 3-cycle through each vertex of a labeled vertex set. We end by illustrating their use in the uniform sampling of simple directed graph realizations from a fixed degree sequence.

We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and mean-field theories of spreading processes. Its structure is well understood for large, sparse, simple graphs: the expected number of edges between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many network data sets are neither large, sparse, nor simple, and in these cases the standard approximation no longer applies. We derive a novel estimator using a dynamical approach: the estimator emerges from the stationarity conditions of a class of Markov Chain Monte Carlo algorithms for graph sampling. We derive error bounds for this estimator and provide an efficient scheme with which to compute it. We test the estimator on synthetic and empirical degree sequences, finding that it enjoys relative error against ground truth a full order of magnitude smaller than the standard approximation. We then compare modularity maximization techniques using both the standard and novel estimators, finding that the qualitative structure of the optimization landscape depends significantly on the estimator choice. Our results emphasize the importance of using carefully specified random graph models in data scientific applications.

In this paper, we investigate the structures of extremal trees which have the minimal number of subtrees in the set of all trees with a given degree sequence. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $${\pi = (d_1, \ldots, d_{5}, 1, \ldots, 1)}$$ ? = ( d 1 , ? , d 5 , 1 , ? , 1 ) have been characterized.

This paper is devoted to the proof of the surprising fact that almost all trees have an even number of independent vertex subsets (in the sense that the proportion of those trees with an odd number of independent sets tends to $0$ as the number of vertices approaches $\infty$) and to its generalisation to other moduli: for fixed $m$, the probability that a randomly chosen tree on $n$ vertices has a number of independent subsets that is divisible by $m$ tends to $1$ as $n \to \infty$.

Random graph models can help us assess the significance of the structural properties of real complex systems. Given the value of a graph property and its value in a randomized ensemble, we can determine whether the property is explained by chance by comparing its real value to its value in the ensemble. The conclusions drawn with this approach obviously depend on the choice of randomization. We argue that keeping graphs in one connected piece, or component, is key for many applications where complex graphs are assumed to be connected either by definition (e.g. the Internet) or by construction (e.g. a crawled subset of the World-Wide Web obtained only by following hyperlinks). Using an heuristic to quickly sample the ensemble of small connected simple graphs with a fixed degree sequence, we investigate the significance of the structural patterns found in real connected graphs. We find that, in sparse networks, the connectedness constraint changes degree correlations, the outcome of community detection with modularity, and the predictions of percolation on the ensemble.

On subtrees of trees

Let ω(G) be the clique number of a graph G. We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values ω(G) completely cover a line segment [a,b] of positive integers. For an arbitrary graphic degree sequence d, we define min(ω,d) and max(ω,d) as follows: $$$$ where \(\) is the graph of realizations of d.

We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values χ(G) of the function chromatic number completely cover a line segment [a,b] of positive integers. Thus for an arbitrary graphical sequence d, two invariants minχ(d):=a and maxχ(d):=b naturally arise. For a regular graphical sequence d=rn:=(r,r,…,r) where r is the degree and n is the number of vertices, the exact values of a and b are found in all situations, except the case where n and r are both even and n<2r.

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas. randomly permute all $2^k\binom{n}{k}$ possible clauses over the variables x1,. . .,xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if, after its addition, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order).Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties have been studied, such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting.Our main contribution is as follows. For m ≥ cn, c = c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e−Ω(m/n)n of the variables take the same value in all satisfying assignments. We also describe a polynomial-time algorithm that finds w.h.p. a satisfying assignment for such formulas.

On [formula omitted]-span and [formula omitted]-span of trees and full binary trees

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

The advent of fuzzy sets, and consequently fuzzy graphs, has solved many problems in ambiguous and uncertain contexts. It is interesting and necessary to study the Wiener index in a cubic fuzzy graph that employs both fuzzy membership and interval-valued fuzzy membership at the same time. In this paper, the Wiener index in a cubic fuzzy graph is introduced as a cubic fuzzy number and some related results are described. The comparison between connectivity index and Wiener index, changes in Wiener index through deleting a node or an edge, and determining the Wiener index in some specific cubic fuzzy graphs have been the other topics studied in this research. In addition, the Wiener index is determined by mentioning concepts of the saturated cubic fuzzy cycle. In this review, the Wiener index is shown as a combination of classical and interval numbers. The results indicate that when some vertices are removed, the Wiener index may change. However, this change will not be exclusively related to both values. Finally, an application of the Wiener index is presented in the study of the properties of some monomer molecules.

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.