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Full-Text Articles in Mathematics
A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi
A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi
Thomas Laurent
The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities.” One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques …
Strong Chromatic Index Of Subset Graphs, Jennifer Quinn, Arthur Benjamin
Strong Chromatic Index Of Subset Graphs, Jennifer Quinn, Arthur Benjamin
Jennifer J. Quinn
The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph Sm(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that sq(Sm(k, l) ) = m choose l-k and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, …
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Noah Prince
A Roman dominating function of a graph G is a labeling f: V(G) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ʋϵV(G)f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(G) + …
Highly Connected Monochromatic Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Monochromatic Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Noah Prince
We consider the following question of Bollobás: given an r-coloring of E(Kn), how large a k-connected subgraph can we find using at most s colors? We provide a partial solution to this problem when s=1 (and n is not too small), showing that when r=2 the answer is n−2k+2, when r=3 the answer is ⌊(n−k)/2⌋+1 or ⌈(n−k)/2⌉+1, and when r−1 is a prime power then the answer lies between n/(r−1)−11(k2−k)r and (n−k+1)/(r−1)+r. The case s≥2 is considered in a subsequent paper (Liu et al.[6]), where we also discuss some of the more glaring open problems relating to this question.
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Hua Kun Liu
Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
Noah Prince
Erdös and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Noah Prince
Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
Noah Prince
Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and …