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Learning a suitable graph is an important precursor to many graph signal processing (GSP) pipelines, such as graph spectral signal compression and denoising. Previous graph learning algorithms either i) make some assumptions on connectivity…

This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or…

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of…

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…

The concept of the integrated adjacency matrix for mixed graphs was first introduced in [9], where its spectral properties were analyzed in relation to the structural characteristics of the mixed graph. Building upon this foundation, this…

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get…

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…

The spectrum of a graph $G$ is the set of the eigenvalues of its adjacency matrix. It turns out that one can say a lot about a graph with the only knowledge being the spectrum of this graph. In this paper we obtain new results about the…

We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le…

Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $\alpha\in[0,1]$, the generalized distance matrix…

In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a…

An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of…

We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…

Let A be a graph type and B an equivalence relation on a group $G$. Let $[g]$ be the equivalence class of $g$ with respect to the equivalence relation B. The B superA graph of $G$ is an undirected graph whose vertex set is $G$ and two…

For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to…

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the…