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 $c > 0$

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

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 ${\rm diam}(G[A])$

Reconstruction of Trees from Jumbled and Weighted Subtrees

Let T be an edge-labeled graph, where the labels are from a finite alphabet Sigma. For a subtree U of T the Parikh vector of U is a vector of length |Sigma| which specifies the multiplicity of each label in U. We ask when T can be reconstructed from the multiset of Parikh vectors of all its subtrees, or all of its paths, or all of its maximal paths. We consider the analogous problems for weighted trees. We show how several well-known reconstruction problems on labeled strings, weighted strings and point sets on a line can be included in this framework. We present reconstruction algorithms and non-reconstructibility results, and extend the polynomial method, previously applied to jumbled strings [Acharya et al., SIAM J. on Discr. Math, 2015] and weighted strings [Bansal et al., CPM 2004], to deal with general trees and special tree classes

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

Reconstruction of Trees from Jumbled and Weighted Subtrees

Let T be an edge-labeled graph, where the labels are from a finite alphabet Sigma. For a subtree U of T the Parikh vector of U is a vector of length |Sigma| which specifies the multiplicity of each label in U. We ask when T can be reconstructed from the multiset of Parikh vectors of all its subtrees, or all of its paths, or all of its maximal paths. We consider the analogous problems for weighted trees. We show how several well-known reconstruction problems on labeled strings, weighted strings and point sets on a line can be included in this framework. We present reconstruction algorithms and non-reconstructibility results, and extend the polynomial method, previously applied to jumbled strings [Acharya et al., SIAM J. on Discr. Math, 2015] and weighted strings [Bansal et al., CPM 2004], to deal with general trees and special tree classes

A Study on Graph Theory of Path Graphs

A simple graph G = (V, E) consists of V , a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. Simple graphs have their limits in modeling the real world. Instead, we use multigraphs, which consist of vertices and undirected edges between these vertices, with multiple edges between pairs of vertices allowed. In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order v1, v2, …, vn such that the edges are {vi, vi+1} where i = 1, 2, …, n ? 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest

Some Topics of Special Interest in Graph Theory

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Strongly indexable graphs

AbstractA (p, q)-graph G = (V, E) is said to be strongly k-indexable if it admits a strong k-indexer viz., an injective function ƒ:V→{0, 1, 2, …, p − 1} such that ƒ(x)+ƒ(y)=ƒ+(xy)ϵ⨍+(E)={k, k + 1, k + 2, …, k + q − 1}.In the terms defined here, k will be omitted if it happens to be unity. We find that a strongly indexable graph has exactly one nontrivial component which is either a star or has a traingle. In any strongly k-indexable graph the minimum point degree is at most 3. Using this fact we show that there are exactly three strongly indexable regular graphs, viz. K2, K3 and K2xK3. If an eulerian (p, q)-graph is strongly indexable then q ϵ 0, 3(mod4)