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We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization…
Graph theory has drawn a lot of attention in the field of Neuroscience during the last decade, mainly due to the abundance of tools that it provides to explore the interactions of elements in a complex network like the brain. The local and…
Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us…
Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity,…
This paper proposes a new graph proximity measure. This measure is a derivative of network reliability. By analyzing its properties and comparing it against other proximity measures through graph examples, we demonstrate that it is more…
Similarity measures are used extensively in machine learning and data science algorithms. The newly proposed graph Relative Hausdorff (RH) distance is a lightweight yet nuanced similarity measure for quantifying the closeness of two graphs.…
A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbes on the oriented edges such that they get conjugated when we reverse the orientation. In this paper we…
We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can…
Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the…
To expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with…
We define a new measure of network symmetry that is capable of capturing approximate global symmetries of networks. We apply this measure to different networks sampled from several classic network models, as well as several real-world…
We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families…
A wide range of tasks in network analysis, such as clustering network populations or identifying anomalies in temporal graph streams, require a measure of the similarity between two graphs. To provide a meaningful data summary for…
Large-scale graphs are widely used to represent object relationships in many real world applications. The occurrence of large-scale graphs presents significant computational challenges to process, analyze, and extract information. Graph…
Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in…
Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$…
We introduce new distance measures for comparing straight-line embedded graphs based on the Fr\'echet distance and the weak Fr\'echet distance. These graph distances are defined using continuous mappings and thus take the combinatorial…
Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ.…
We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has…
A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two…