Related papers: Enumerating simple paths from connected induced su…
A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that…
The $t$-connected ideal of a graph $G$ is generated by all connected induced subgraphs of $G$ with $t$ vertices. When $t = 2$, this coincides with the usual edge ideal of the graph. Following the work of Faridi et al., we give a…
This paper studies induced paths in strongly regular graphs. We give an elementary proof that a strongly regular graph contains a path $P_4$ as an induced subgraph if and only if it is primitive, i.e. it is neither a complete multipartite…
We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to…
We propose a universal approach to a range of enumeration problems in graphs. The key point is in contracting suitably chosen symmetric tensors placed at the vertices of a graph along the edges. In particular, this leads to an algorithm…
In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the…
Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…
The pathwidth of a graph $G$ is the smallest $w\in \mathbb{N}$ such that $G$ can be constructed from a sequence of graphs, each on at most $w+1$ vertices, by gluing them together in a linear fashion. We provide a full classification of the…
In this work we consider the Laplacian controllability of a graph constructed by interconnecting a finite number of single-input Laplacian controllable graphs. We first study the interconnection realized by the composite graph of two…
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…
A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the…
A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates…
We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of…
The Hamiltonian Path Problem is formulated as a continuous minimization problem on conductances assigned to an Ohmic network associated with the given graph. The objective function is a sum of two penalty terms that collectively enforce a…
We define the crossing graph of a given embedded graph (such as a road network) to be a graph with a vertex for each edge of the embedding, with two crossing graph vertices adjacent when the corresponding two edges of the embedding cross…
A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$…
An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path…
We prove that among all graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees,…