Largest Laplacian Eigenvalue Research Articles

Sharp lower bounds for the Laplacian Estrada index of graphs

Let G be a simple graph on n vertices, and let λ 1 , λ 2 , … , λ n be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE ( G ) = ∑ i = 1 n e λ i . Consider a graph G with n ≥ 3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue λ n . Let K n , S n , and K p , q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE ( G ) ≥ n e 2 m n + c + e λ n − ( c + 1 ) e λ n c + 1 . Furthermore, we show that the equality holds if and only if G ≅ K ¯ n (the complement of K n ), G ≅ ∪ i = 1 c − 1 K 1 ∪ S c + 1 if n = 2c, or G ≅ K n 2 , n 2 if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175–180.] is not complete. Consequently, we provide a complete version of this lower bound.

Brouwer's Conjecture for the cartesian product of graphs

Let G be a simple, undirected graph with n vertices and m edges and Sk(G) the sum of its k largest Laplacian eigenvalues. Brouwer conjectured that Sk(G)≤m+(k+12) for any 1≤k≤n. We show that if G and H are simple undirected graphs satisfying Brouwer's Conjecture, then the cartesian product G□H also satisfies Brouwer's Conjecture.

On the Ky Fan $k$-norm of the $LI$-matrix of graphs

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix of $G$ is defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, which is closely related to the well known problems and results in spectral graph theory, such as the Laplacian spectral radius, the Laplacian spread, the sum of the $k$ largest Laplacian eigenvalues, the Laplacian energy, and other parameters. Some bounds on the Ky Fan $k$-norm of the $LI$-matrix of graphs are given, and the extremal graphs are partly characterized. In addition, upper and lower bounds on the Ky Fan $k$-norm of $LI$-matrix of trees, unicyclic graphs, and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.

Network Similarity Measure and Ediz Eccentric Connectivity Index

Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.

Connective Steiner 3-eccentricity index and network similarity measure

Abstract For a set S⊆V(G) in a network G, the Steiner distance dG(S) of S is the minimum size among all connected subnetworks whose vertex sets contain S. The Steiner k-eccentricity ɛk(v) of a vertex v of G is the maximum Steiner distance among all k-vertex set S which contains the vertex v, i.e., e k ( v ) = max { d ( S ) | S ⊆ V ( G ) , | S | = k , v ∈ S } . Based on Steiner k-eccentricity, the connective Steiner k-eccentricity index is introduced. As a newly structural invariant, some properties of the connective Steiner 3-eccentricity index are investigated. Firstly we present an O(n2)-polynomial time algorithm to calculate the connective Steiner 3-eccentricity index of trees. Secondly some optimal problems among some network classes are discussed. As its application, finally we consider the network similarity measure based on the connective Steiner 3-eccentricity index. By two different methods, we study its advantages. Numerical results show that the measure based on the connective Steiner 3-eccentricity index has more advantages than the ones based on other topological indices (graph energy, Randic index, the largest adjacent eigenvalue, the largest Laplacian eigenvalue).

On the sum of signless Laplacian spectra of graphs

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.

Nordhaus–Gaddum type inequalities for the two largest Laplacian eigenvalues

For a simple graph G with n vertices, let μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0 be the Laplacian eigenvalues of G. We study Nordhaus–Gaddum type inequalities for μ1(G) and μ2(G). We improve some existing results from the literature for μ1(G) and obtain new results for μ2(G). We also propose a conjecture that μ2(G)+μ2(G¯)≤2n−2holds for any graph G and prove it is true when G or G¯ is disconnected; when G is bipartite; for regular graphs and when G and G¯ have diameter not equal to 2.

On a Lower Bound for the Laplacian Eigenvalues of a Graph

If $\mu_m$ and $d_m$ denote, respectively, the $m$-th largest Laplacian eigenvalue and the $m$-th largest vertex degree of a graph, then $\mu_m \geqslant d_m-m+2$. This inequality was conjectured by Guo in 2007 and proved by Brouwer and Haemers in 2008. Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying $\mu_m = d_m-m+2$. In particular we give a full classification of graphs with $\mu_m = d_m-m+2 \leqslant 1$.

Spectral properties of Pascal graphs

We obtain spectral properties of the Pascal graphs by exploring its spectral graph invariants such as the algebraic connectivity, the first three largest Laplacian eigenvalues and the nullity. Some open problems pertaining to the Pascal graphs are given.

Bicyclic graphs with maximum sum of the two largest Laplacian eigenvalues

Let G be a simple connected graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. In this paper, we determine the bicyclic graph with maximum $S_{2}(G)$ among all bicyclic graphs of order n, which confirms the conjecture of Guan et al. (J. Inequal. Appl. 2014:242, 2014) for the case of bicyclic graphs.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let Hk[G] be the lexicographic product of Hk and G, where Hk is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of Hk[G] and Hk when G and H are regular and the Laplacian spectrum of Hk[G] and Hk for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

On signed graphs whose second largest Laplacian eigenvalue does not exceed 3

Let be a signed graph, where G is the underlying graph and is the signature function on the edges of G. In this paper, we consider the Laplacian eigenvalues of signed graphs and we characterize the connected signed graphs whose second largest Laplacian eigenvalue does not exceed 3. Furthermore, we study the Laplacian spectral determination of most graphs in the latter family.

On the sum of the two largest Laplacian eigenvalues of unicyclic graphs

Let G be a simple graph and $S_{2}(G)$ be the sum of the two largest Laplacian eigenvalues of G. Guan et al. (J. Inequal. Appl. 2014:242, 2014) determined the largest value for $S_{2}(T)$ among all trees of order n. They also conjectured that among all connected graphs of order n with m ( $n \leq m\leq2n -3$ ) edges, $G_{m,n}$ is the unique graph which has maximal value of $S_{2}(G)$ , where $G_{m,n}$ is a graph of order n with m edges which has $m-n+1$ triangles with a common edge and $2n-m-3$ pendent edges incident with one end vertex of the common edge. In this paper, we confirm the conjecture with $m=n$ .

On limit points of the third largest Laplacian eigenvalues of graphs

On limit points of the third largest Laplacian eigenvalues of graphs

On the sum of the two largest Laplacian eigenvalues of trees

Abstract For S ( T ) , the sum of the two largest Laplacian eigenvalues of a tree T, an upper bound is obtained. Moreover, among all trees with n ≥ 4 vertices, the unique tree which attains the maximal value of S ( T ) is determined. MSC:05C50.

Graphs with small second largest Laplacian eigenvalue

Let L(G) be the Laplacian matrix of G. In this paper, we characterize all of the connected graphs with second largest Laplacian eigenvalue no more than l; where l≐3.2470 is the largest root of the equation μ3−5μ2+6μ−1=0. Moreover, this result is used to characterize all connected graphs with second largest Laplacian eigenvalue no more than three.

On matrices associated to directed graphs and applications

This paper deals with the notions of 0-incidence and 1-incidence between edges on a directed graph associated to the line graph of a graph. The Laplacian energy and the signless Laplacian energy are obtained in a new way. From these results a relation between both energies is derived. Moreover, we obtain lower bounds for both the largest Laplacian eigenvalue and the largest signless Laplacian eigenvalue and prove that the latter is strictly greater than the first one.

Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Let be weighted graphs, as the graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of Laplacian matrix of a graph . We obtain two upper bounds for the largest Laplacian eigenvalue of weighted graphs and we compare these bounds with previously known bounds.

Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. LetG=(V,E)be a simple connected graph onnvertices and letμ(G)be the largest Laplacian eigenvalue (i.e., the spectral radius) ofG. In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius ofG.

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.