In which we introduce the Leighton-Rao relaxation of sparsest cut. Let G = (V, E) be an undirected graph. Unlike past lectures, we will not need to assume that G is regular. We are interested in finding a sparsest cut in G, where the sparsity of a non-trivial bipartition (S, V − S) of the vertices is which is the ratio between the fraction of edges that are cut by (S, V − S) and the fraction of...

A balanced bipartition of a graph G is a bipartition V1 and V2 of V (G) such that −1 ≤ |V1| − |V2| ≤ 1. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V1, V2 such that max{e(V1), e(V2)} ≤ m/3, where e(Vi) denotes the number of edges of G with both ends in Vi. In this note, we prove this conjecture for graphs wi...

Topologically-ordered phases of matter at nonzero temperature are conjectured to exhibit universal patterns long-range entanglement, which can be detected using the entanglement negativity, a mixed-state measure. In this paper, we show that negativity in certain topological orders understood through properties an emergent symmetry-protected (SPT) order is localized on bipartition. This connecti...

We present an algorithm which partitions a data set in two parts with equal size and experimentally nearly the same distribution measured through the likelihood of a Parzen kernel density estimator. The generation of the partition takes O( 1 2 N(N − 1)) operations (N number of data) and is 2 orders of magnitude faster than the state of the art. 1 Generating an equi-distributed bipartition 1.1 P...

‎A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$‎. ‎The total domination number of a graph $G$‎, ‎denoted by $gamma_t(G)$‎, ‎is~the minimum cardinality of a total dominating set of $G$‎. ‎Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International ournal of Graphs and Combinatorics 1 (2004)‎, ‎6...

The Tree of Life has provided a framework to study the evolution of organisms. Phylogenetic trees are used to depict the evolution of organisms or of molecules. Comparative genome analyses have shown that genomes are mosaics where different parts have different histories. This discovery calls into question the notion of a unified evolutionary tree for an organism and gives rise to the notion of...

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a ra...

We prove that every tree of order n 5 with three leaves is (n + 1)-unavoidable. More precisely, we prove that every tree A with three leaves of order n is contained in every tournament T of order n + 1 except if (T ; A) is (R 5 ; S + 3) or its dual, where R 5 is the regular tournament on ve vertices and S + 3 is the outstar of degree three, that is, the tree consisting of a root dominating thre...

For the terminology and notation we follow Bondy and Murty [1] in graph theory and Welsh [3] in matroid theory. All sets considered are finite. Let G be a bipartite graph with bipartition (X, Y) and let V and E denote, respectively, the vertex and edge sets of G. A matching M in G is called a matching of X into Y if every vertex in X is M-saturated. If S is any subset of V, we shall write <5(S)...