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In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications…
The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are…
A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…
We propose two-channel critically-sampled filter banks for signals on undirected graphs that utilize spectral domain sampling. Unlike conventional approaches based on vertex domain sampling, our transforms have the following desirable…
A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. Let $\mathcal{B}(\mathbf{k}, \alpha)$ be the class of bi-block graph on $\mathbf{k}$ vertices with given independence number $\alpha$. It is…
We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited…
The double graph of a graph $G$ is defined as $\mathcal{D}[G]$ = $G \times T_2$, where \(T_2\) is the total graph with 2 vertices and $\times$ stands for the Kronecker product of graphs. In this paper, sufficient conditions for double…
In this paper, motivated by a question posed in \cite{AH}, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove…
Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed…
An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked…
The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257]…
It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph…
We initiate the study of deterministic distributed graph algorithms with predictions in synchronous message passing systems. The process at each node in the graph is given a prediction, which is some extra information about the problem…
A perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a…
The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. The profound relationship between a planar graph and its dual has been the algorithmic basis…
Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal…
Recently, one has seen a surge of interest in developing such methods including ones for learning such representations for (undirected) graphs (while preserving important properties). However, most of the work to date on embedding graphs…
In this paper, matching pairs of random graphs under the community structure model is considered. The problem emerges naturally in various applications such as privacy, image processing and DNA sequencing. A pair of randomly generated…