Let $G$ be a simple graph and $α$ a real number. The quantity $s_α(G)$ defined as the sum of the $α$-th power of the non-zero Laplacian eigenvalues of $G$ generalizes several concepts in the literature. The Laplacian Estrada index is a newly introduced graph invariant based on Laplacian eigenvalues. We establish bounds for $s_α$ and Laplacian Estrada index related to the degree sequences.
We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.
The signless Laplacian Estrada index of a graph $G$ is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we show that there are exactly two tricyclic graphs with the maximal signless Laplacian Estrada index.
Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.
For a simple and connected graph, several lower and upper bounds of graph invariants expressed in terms of the eigenvalues of the normalized Laplacian matrix have been proposed in literature. In this paper, through a unified approach based on majorization techniques, we provide some novel inequalities depending on additional information on the localization of the eigenvalues of the normalized Laplacian matrix. Some numerical examples show how sharper results can be obtained with respect to those existing in literature.
Let $G$ be a graph on $n$ vertices and $λ_1,λ_2,\ldots,λ_n$ its eigenvalues. The Estrada index of $G$ is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present some new lower bounds obtained for the Estrada Index of graphs and in particular of bipartite graphs that only depend on the number of vertices, the number of edges, Randić index, maximum and minimum degree and diameter.
Let $λ_1,λ_2,\cdots,λ_n$ be the eigenvalues of the distance matrix of a connected graph $G$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{λ_i}$. In this note, we present new lower and upper bounds for $DEE(G)$. In addition, a Nordhaus-Gaddum type inequality for $DEE(G)$ is given.
Let $G$ be a graph on $n$ vertices and $λ_1\geq λ_2\geq \ldots \geq λ_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{λ_i}.$ In this work, we using an increasing sequence converging to the $λ_1$ to obtain an increasing sequence of lower bounds for $EE(G)$. In addition, we generalize this succession for the Estrada index of an arbitrary nonnegative Hermitian matrix.
For a simple, undirected and connected graph $G$, $D_α(G) = αTr(G) + (1-α) D(G)$ is called the $α$-distance matrix of $G$, where $α\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $α$-distance energy of $G$ was defined based on the spectra of $D_α(G)$. In this paper, we define the $α$-distance Estrada index of $G$ in terms of the eigenvalues of $D_α(G)$. And we give some bounds on the spectral radius of $D_α(G)$, $α$-distance energy and $α$-distance Estrada index of $G$.
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.
In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than $1$ are simple and also the multiplicity of Laplacian eigenvalue $1$ has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order $n$ that have a multiplicity that is close to the upper bound $\frac{n-3}{2}$, and emphasize the particular role of the algebraic connectivity.
We show that the number of Laplacian eigenvalues greater than the average degree of a tree having $n$ vertices is at most $\lfloor\frac{n}{2} \rfloor$.
Let $G$ be a connected simple graph with $n$ vertices. 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 $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$. The largest eigenvalue $\partial_{1}^{L}(G)$ is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized …
Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over $\mathbb{Z}_m$ or $\mathbb{Z}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signle…
We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $ρ_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $ν(G)$ the independence number of $G$. Recently, the paper [Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, arXiv:1912.13227] discussed the graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$. However, there is one remaining case (graphs with $ρ_{n-1}(G)\neq 1$ and $ν(G)= 2$) not considered. In this paper, we focus on cographs and graphs wi…
Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.
We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, in the limit $p\to 1$ upper and lower bounds coincide.
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let $ρ_{n-1}(G)$ and $ν(G)$ be the second least normalized Laplacian eigenvalue and the independence number of a graph $G$, respectively. As the main conclusions, two families of $n$-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$ are determined: graphs with $ρ_{n-1}(G)=-1$ and graphs with $ρ…
Let $G$ be a connected simple graph of order $n$. Let $ρ_1(G)\geq ρ_2(G)\geq \cdots \geq ρ_{n-1}(G)> ρ_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(ρ_i)$ the multiplicity of the normalized Laplacian eigenvalue $ρ_i$. Let $ν(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365…
The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalu…
Motivated by classic tree algorithms, in 1995 we designed a bottom-up $O(n)$ algorithm to compute the determinant of a tree's adjacency matrix $A$. In 2010 an $O(n)$ algorithm was found for constructing a diagonal matrix congruent to $A + xI_n$, $x \in \mathbb{R}$, enabling one to easily count the number of eigenvalues in any interval. A variation of the algorithm allows Laplacian eigenvalues in trees to be counted. We conjecture that for any tree $T$ of order $n \geq 2$, at least half of its Laplacian eigenvalues are less than $\bar{d} = 2 - \frac{2}{n}$, its average vertex degree.
In this paper, we establish the relation between classic invariants of graphs and their integer Laplacian eigenvalues, focusing on a subclass of chordal graphs, the strictly chordal graphs, and pointing out how their computation can be efficiently implemented. Firstly we review results concerning general graphs showing that the number of universal vertices and the degree of false and true twins provide integer Laplacian eigenvalues and their multiplicities. Afterwards, we prove that many integer Laplacian eigenvalues of a strictly chordal graph are directly related to particular simplicial ver…
In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing for them some new properties.
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the weighted case.
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
