Related papers: Operations on Graphs Increasing Some Graph Paramet…
We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard…
We propose the notion of normalized Laplacian matrix $\mathcal{L}(\Phi)$ for a gain graphs and study its properties in detail, providing insights and counterexamples along the way. We establish bounds for the eigenvalues of…
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph…
We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary…
This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as…
Throughout this work, the vertex decomposability and shellability of graphs formed from other graphs by various operations are investigated. Also among the other things, by using some graph operations, new classes of Cohen-Macaulay graphs…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations…
To each graph on $n$ vertices there is an associated subspace of the $n \times n$ matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally…
The permanent of an $n \times n$ matrix $M = (m_{ij})$ is defined as $\mathrm{per}(M) = \sum_{\sigma \in S_n} \prod_{i=1}^n m_{i,\sigma(i)}$, where $S_n$ denotes the symmetric group on $\{1,2,\ldots,n\}$. The permanental polynomial of $M$,…
We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $\gamma$-polynomials of their associated graph-associahedra. The partial…
The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful…
A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their…
Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of…
Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…
We consider the problem of controlling a partially-observed dynamic process on a graph by a limited number of interventions. This problem naturally arises in contexts such as scheduling virus tests to curb an epidemic; targeted marketing in…
In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and…
We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators,…
Theoretical analyses for graph learning methods often assume a complete observation of the input graph. Such an assumption might not be useful for handling any-size graphs due to the scalability issues in practice. In this work, we develop…
Graph neural networks process information on graphs represented at a given resolution scale. We analyze the effect of using different coarse-grained graph resolutions, obtained through the Laplacian renormalization group theory, on node…