Related papers: On graphs with three or four distinct normalized L…
We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph…
This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…
The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…
In this paper, we determine the two normalized Laplacian spectrum of generalized subdivision- vertex corona, subdivision-edge corona for a connected regular graph with an arbitrary reg- ular graph in terms of their normalized Laplacian…
A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified
The graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs $\mathcal{L}^k(G)$ with all equal negative eigenvalues $-2$ are characterized for $k\ge…
We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every…
We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the…
A constructive characterization of the class of uniformly $4$-connected graphs is presented. The characterization is based on the application of graph operations to appropriate vertex and edge sets in uniformly $4$-connected graphs, that…
This paper explores interlacing inequalities in the Laplacian spectrum of signed cycles and investigates interlacing relationship between the spectrum of the net-Laplacian of a signed graph and its subgraph formed by removing a vertex…
In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2\Delta(G)$, where $\Delta(G)$ is the largest degree of $G$, if and only if $G$ is…
Let $G$ be a connected graph of order $n$ with girth $g$. For $k=1,\dots,\min\{g-1, n-g\}$, let $n(G,k)$ be the number of Laplacian eigenvalues (counting multiplicities) of $G$ that fall inside the interval $[n-g-k+4,n]$. We prove that if…
A number of recent papers have considered signed graph Laplacians, a generalization of the classical graph Laplacian, where the edge weights are allowed to take either sign. In the classical case, where the edge weights are all positive,…
In this note, we present a structural description of certain connected cographs having $k \geq 2$ main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are quasi-threshold graphs with two main…
Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be…
The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest…
Let $G$ be a unicyclic graph. In this paper, we provide an upper bound for the number of Laplacian eigenvalues of $G$ within the interval $[0,1)$ in terms of the diameter and the girth of $G$.
To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two…
The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. First, a class of directed signed graphs is investigated in which one pair of nodes (either connected or not) is…
The set $S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,n-1,n\}\setminus\{i,j\},\quad 0<i<j\leqslant n$, is called Laplacian realizable if there exists a simple connected graph $G$ whose Laplacian spectrum is $S_{\{i,j\}_{n}^{m}}$.…