Connected Graphs with a Given Degree Sequence: Efficient Sampling, Correlations, Community Detection and Robustness

2020 ◽  
pp. 33-47 ◽  
Author(s):  
John H. Ring ◽  
Jean-Gabriel Young ◽  
Laurent Hébert-Dufresne
Keyword(s):  
Community Detection ◽  
Degree Sequence ◽  
Connected Graphs ◽  
Efficient Sampling

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

scholarly journals Per-Spectral Characterizations of Bicyclic Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Tingzeng Wu ◽  
Huazhong Lü
Keyword(s):  
Community Detection ◽  
Connected Graphs ◽  
Spectral Techniques ◽  
Network Properties ◽  
Spectral Characterizations

Spectral techniques are used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, and so forth. In this paper, we investigate which kinds of bicyclic networks are determined by their per-spectra. We find that the permanental spectra cannot determine sandglass graphs in general. When we restrict our consideration to connected graphs or quadrangle-free graphs, sandglass graphs are determined by their permanental spectra. Furthermore, we construct countless pairs of per-cospectra bicyclic networks.

scholarly journals Edge-Decomposition of Graphs into Copies of a Tree with Four Edges

10.37236/2110 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
János Barát ◽  
Dániel Gerbner
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Degree Sequence ◽  
Decomposition Theorem ◽  
Bipartite Graphs ◽  
Connected Graphs ◽  
Edge Decomposition

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013).Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

scholarly journals Critical Window for Connectivity in the Configuration Model

2017 ◽  
Vol 26 (5) ◽  
pp. 660-680 ◽  
Author(s):  
LORENZO FEDERICO ◽  
REMCO VAN DER HOFSTAD
Keyword(s):  
Degree Sequence ◽  
Random Variable ◽  
Giant Component ◽  
Configuration Model ◽  
Moment Condition ◽  
Connected Graphs ◽  
Critical Window ◽  
Connectivity Probability ◽  
Asymptotic Number ◽  
Asymptotic Probability

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

scholarly journals Graphs with Given Degree Sequence and Maximal Spectral Radius

10.37236/843 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Türker Bıyıkoğlu ◽  
Josef Leydold
Keyword(s):  
Spectral Radius ◽  
Degree Sequence ◽  
Breadth First Search ◽  
Connected Graphs

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization.

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