Related papers: Some inner metric parameters of a digraph: Iterate…
In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral…
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We…
Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the "AIM Minimum Rank -- Special Graphs Work Group", whereas metric dimension is a well-known graph parameter. We…
A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant…
We study the ratio, in a finite graph, of the sizes of the largest matching in any pair of disjoint matchings with the maximum total number of edges and the largest possible matching. Previously, it was shown that this ratio is between 4/5…
This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…
This article investigates the connectivity dimension of a graph. We introduce this concept in analogy to the metric dimension of a graph, providing a graph parameter that measures the heterogeneity of the connectivity structure of a graph.…
This article studies the zero divisor graphs of the ring of Lipschitz integers modulo $n$. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these…
Directed acyclic graphs whose nodes are all the divisors of a positive integer $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered. Fourteen graph invariants such as order, size, and the number of paths are investigated for two…
Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…
In a metric space $M=(X,d)$, we say that $v$ is between $u$ and $w$ if $d(u,w)=d(u,v)+d(v,w)$. Taking all triples $\{u,v,w\}$ such that $v$ is between $u$ and $w$, one can associate a 3-uniform hypergraph with each finite metric space $M$.…
Supervised learning on graphs is a challenging task due to the high dimensionality and inherent structural dependencies in the data, where each edge depends on a pair of vertices. Existing conventional methods are designed for standard…
In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$…
The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. A minimizer graph is such that minimizes the spectral radius among all connected graphs on $n$ vertices with diameter $d$. The minimizer graphs are known for…
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean…
The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. The theoretical properties of the edge metric representations and the edge metric dimension of generalized Petersen graphs…
Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The…
Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative…