Homometric sets in trees

Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least $cn^{2/3}$ for a constant $c > 0$

Subsets of vertices of the same size and the same maximum distance

For a simple connected graph $G=(V,E)$ and a subset $X$ of its vertices, let $$d^*(X) = \max\{{\rm dist}_G(x,y): x,y\in X\}$$ and let $h^*(G)$ be the largest $k$ such that there are disjoint vertex subsets $A$ and $B$ of $G$, each of size $k$ such that $d^*(A) = d^*(B).$ Let $h^*(n) = \min \{h^*(G): |V(G)|=n\}$. We prove that $h^*(n) = \lfloor (n+1)/3 \rfloor,$ for $n\geq 6.$ This solves the homometric set problem restricted to the largest distance exactly. In addition we compare $h^*(G)$ with a respective function $h_{{\rm diam}}(G)$, where $d^*(A)$ is replaced with ${\rm diam}(G[A])$

Homometric Number of Graphs

Given a graph G=(V,E), two subsets S_1 and S_2 of the vertex set V are homometric, if their distance multi sets are equal. The homometric number h(G) of a graph G is the largest integer k such that there exist two disjoint homometric subsets of cardinality k. We find lower bounds for the homometric number of the Mycielskian of a graph and the join and the lexicographic product of two graphs. We also obtain the homometric number of the double graph of a graph, the cartesian product of any graph with K_2 and the complete bipartite graph. We also introduce a new concept called weak homometric number and find weak homometric number of some graphs

Twin subgraphs and core-semiperiphery-periphery structures

A standard approach to reduce the complexity of very large networks is to group together sets of nodes into clusters according to some criterion which reflects certain structural properties of the network. Beyond the well-known modularity measures defining communities, there are criteria based on the existence of similar or identical connection patterns of a node or sets of nodes to the remainder of the network. A key notion in this context is that of structurally equivalent or twin nodes, displaying exactly the same connection pattern to the remainder of the network. The first goal of this paper is to extend this idea to subgraphs of arbitrary order of a given network, by means of the notions of T-twin and F-twin subgraphs. This is motivated by the need to provide a systematic approach to the analysis of core-semiperiphery-periphery (CSP) structures, a notion which somehow lacks a formal treatment in the literature. The goal is to provide an analytical framework accommodating and extending the idea that the unique (ideal) core-periphery (CP) structure is a 2-partitioned K2. We provide a formal definition of CSP structures in terms of core eccentricities and periphery degrees, with semiperiphery vertices acting as intermediaries. The T-twin and F-twin notions then make it possible to reduce the large number of resulting structures by identifying isomorphic substructures which share the connection pattern to the remainder of the graph, paving the way for the decomposition and enumeration of CSP structures. We compute the resulting CSP structures up to order six. We illustrate the scope of our results by analyzing a subnetwork of the network of 1994 metal manufactures trade. Our approach can be further applied in complex network theory and seems to have many potential extensions

Combinatorics and Probability

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers

Disjoint homometric sets in graphs

Two subsets of vertices in a graph are called homometric if the multisets of distances determined by them are the same. Let h(n) denote the largest number h such that any connected graph of n vertices contains two disjoint homometric subsets of size h. It is shown that (c log n)/(log log n) 3