Laplacian Energy Of Graph Research Articles

On the spectrum, energy and Laplacian energy of graphs with self-loops

The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let Gσ be an order (vertices) ‘p’ graph with ‘q’ edges and σ− self loops. The adjacency matrix of Gσ is defined by A(Gσ)=(aij) if vi∼adjvj then aij=1; if vi=vj where vi∈Vσ then aii=1 and zero otherwise, where Vσ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as E(Gσ)=∑|λi−σ/p|. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior.

Energy of Graph

By given the adjacency matrix, laplacian matrix of a graph we can find the set of eigenvalues of graph in order to discussed about the energy of graph and laplacian energy of graph. (i.e. the sum of eigenvalues of adjacency matrix and laplacian matrix of a graph is called the energy of graph) and the laplacian energy of graph is greater or equal to zero for any graph and is greater than zero for every connected graph with more or two vertices (i.e. the last eigenvalues of laplacian matrix is zero), according to several theorems about the energy of graph and the laplacian energy of graph that are described in this work; I discussed about energy of graph, laplacian energy of graph and comparing them here.

Graph Signal Processing Based Cross-Subject Mental Task Classification Using Multi-Channel EEG Signals

Classification of mental tasks from electroencephalogram (EEG) signals play a crucial role in designing various brain-computer interface (BCI) applications. Most of the current techniques consider each channel as independent, neglecting the functional connectivity of the brain during mental activity and are primarily subject specific. This paper proposes a graph signal representation to classify a pair of mental tasks using multi-channel EEG signals (MTMC-EEG) with cross subject classification within the database. Here, each channel of EEG signal corresponds to nodes of the task based graph whose EEG time series resides on the respective nodes. Functional connectivity of the brain between these nodes is obtained using smoothness constraint based Graph Signal Processing (GSP) technique. Graph spectral features namely, two-norm total variation of eigen vector (TNTV) corresponding to weighted adjacency matrix, graph Laplacian energy (GLE) using eigenvalues of Laplacian matrix and convex sum of TNTV and GLE in the form of joint total variation energy (JTVE) are proposed in this paper. The performance of the proposed methodology is evaluated on publicly available two different databases of MTMC EEG signals using benchmark classifiers and compared with the state of the art. Further, the superiority of the proposed metric obtained from the smoothened graph of GSP technique is validated by comparing it with Pearson correlation and Gaussian radial basis function (RBF) based functional connectivity in terms of accuracy, F-Score, and information transfer rate (ITR). The robustness of the proposed method is validated by adding white Gaussian noise (AWGN) to the EEG signals using different SNRs.

Laplacian Energy of Graphs

Laplacian Energy of Graphs

Minimum dominating signless laplacian graph energy

Minimum dominating graph energy was proposed by Rajesh Kanna et al.[11] in 2013. I propose the Minumum dominating signless laplacian graph energy in this paper. Also obtain some bounds and discuss some important graph classes for the minimum dominating signless Laplacian energy.

On energy and Laplacian energy of chain graphs

Let G be a simple graph of order n. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. The Laplacian energy of the graph G is defined as LE=LE(G)=∑i=1nμi−d¯,where μ1,μ2,…,μn−1,μn=0 are the Laplacian eigenvalues, and d¯ is the average degree of graph G. In this paper we present some lower and upper bounds on E(G) of chain graph G. From this we prove that the star gives the minimal energy of connected chain graphs of order n. We present a lower bound on LE(G) of chain graphs G in terms of order n, and characterize the extremal graphs. Moreover, we obtain the maximal Laplacian energy among all connected chain graphs of order n with n edges. Finally, we propose an open problem on Laplacian energy of chain graphs.

Sharp Bounds on (Generalized) Distance Energy of Graphs

Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

On the sum of the distance signless Laplacian eigenvalues of a graph and some inequalities involving them

The distance signless Laplacian matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. If [Formula: see text] are the distance signless Laplacian eigenvalues of a simple graph [Formula: see text] of order [Formula: see text] then we put forward the graph invariants [Formula: see text] and [Formula: see text] for the sum of [Formula: see text]-largest and the sum of [Formula: see text]-smallest distance signless Laplacian eigenvalues of a graph [Formula: see text], respectively. We obtain lower bounds for the invariants [Formula: see text] and [Formula: see text]. Then, we present some inequalities between the vertex transmissions, distance eigenvalues, distance Laplacian eigenvalues, and distance signless Laplacian eigenvalues of graphs. Finally, we give some new results and bounds for the distance signless Laplacian energy of graphs.

The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.

Minimum covering reciprocal distance signless Laplacian energy of graphs

Abstract Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined as Tr G ′ ( ν ) = ∑ u ∈ V ( G ) 1 d G ( u , ν ) , u ≠ ν . $${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$ The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j), q ij = { 1 + Tr ′ ( ν i ) if i = j and ν i ∈ C Tr ′ ( ν i ) if i = j and ν i ∉ C 1 d ( ν i , ν j ) otherwise $${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} & {{\rm{otherwise}}} & {} & {} & {} \cr } } \right.$$ where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.

On the distance signless Laplacian spectral radius and the distance signless Laplacian energy of graphs

The distance signless Laplacian spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of the distance signless Laplacian matrix of [Formula: see text], defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. In this paper, we determine some bounds on the distance signless Laplacian spectral radius of [Formula: see text] based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.

On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs

Let G be a graph of order n. The energy E(G) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy, the signless Laplacian energy and the distance energy of graph G are denoted by LE(G), SLE(G) and DE(G), respectively. In this paper we introduce a distance Laplacian energy DLE and distance signless Laplacian energy DSLE of a connected graph. We present Nordhaus–Gaddum type bounds on Laplacian energy LE(G) and signless Laplacian energy SLE(G) in terms of order n of graph G and characterize graphs for which these bounds are best possible. The complete graph and the star give the smallest distance signless Laplacian energy DSLE among all the graphs and trees of order n, respectively. We give lower bounds on distance Laplacian energy DLE in terms of n for graphs and trees, and characterize the extremal graphs. Also we obtain some relations between DE, DSLE and DLE of graph G. Moreover, we give several open problems in this paper.

Bounds on Energy and Laplacian Energy of Graphs

Let G be simple graph with n vertices and m edges. The energy E(G) of G, denotedby E(G), is dened to be the sum of the absolute values of the eigenvalues of G. Inthis paper, we present two new upper bounds for energy of a graph, one in terms ofm,n and another in terms of largest absolute eigenvalue and the smallest absoluteeigenvalue. The paper also contains upper bounds for Laplacian energy of graph.

Bounds on Energy and Laplacian Energy of Graphs

Let G be simple graph with n vertices and m edges. The energy E(G) of G, denotedby E(G), is dened to be the sum of the absolute values of the eigenvalues of G. Inthis paper, we present two new upper bounds for energy of a graph, one in terms ofm,n and another in terms of largest absolute eigenvalue and the smallest absoluteeigenvalue. The paper also contains upper bounds for Laplacian energy of graph.

Applications of a theorem by Ky Fan in the theory of weighted Laplacian graph energy

Applications of a theorem by Ky Fan in the theory of weighted Laplacian graph energy

The Laplacian energy of conjugacy class graph of some dihedral groups

Let G be a dihedral group and Gamma its conjugacy class graph. The Laplacian energy of the graph, LE(Gamma) is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the number of vertices. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups and its eigenvalues are first computed. Then, the Laplacian energy of this graph is determined.

Color signless Laplacian energy of graphs

In this paper, we introduce the new concept of color Signless Laplacian energy LEc+(G). It depends on the underlying graph G and the colors of the vertices. Moreover, we compute color signless Laplacian spectrum and the color signless Laplacian energy of families of graph with the minimum number of colors. The color signless Laplacian energy for the complement of some colored graphs is also obtained.

On Energy and Laplacian Energy of Graphs

Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.

On energy, Laplacian energy and p-fold graphs

For a graph $G$ having adjacency spectrum ($A$-spectrum) $\lambda_n\leq\lambda_{n-1}\leq\cdots\leq\lambda_1$ and Laplacian spectrum ($L$-spectrum) $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$, the energy is defined as $ E(G)=\sum_{i=1}^{n}|\lambda_i|$ and the Laplacian energy is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\frac{2m}{n}|$. In this paper, we give upper and lower bounds for the energy of $KK_n^j,~1\leq j \leq n$ and as a consequence we generalize a result of Stevanovic et al. [More on the relation between Energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. {\bf 61} (2009) 395-401]. We also consider strong double graph and strong $p$-fold graph to construct some new families of graphs $G$ for which $E(G)> LE(G)$.

Interactive Segmentation Using Constrained Laplacian Optimization

We present a novel interactive image segmentation approach with user scribbles using constrained Laplacian graph optimization. A novel energy framework is developed by adding the smoothing item in the cost function of Laplacian graph energy. To the best of our knowledge, our approach is the first to incorporate the normalized cuts and graph cuts algorithms into a unified energy optimization framework. The proposed approach is further accelerated by running the proposed optimization method on a band region when we segment the large images. Our acceleration strategy enables our approach to efficiently segment the large images, which yields about a 20-80 times speedup. The proposed approach is evaluated on both the publicly available data sets and our own data set with large images. The benefits of the proposed unified framework are also demonstrated both qualitatively and quantitatively. The experimental results show that our segmentation method achieves better performance of both boundary recall and error rate than the existing state-of-the-art approaches.