scholarly journals More on the Laplacian Estrada index

2009 ◽  
Vol 3 (2) ◽  
pp. 371-378 ◽  
Author(s):  
Bo Zhou ◽  
Ivan Gutman
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
The First Zagreb Index

Let G be a graph with n vertices and let ?1, ?2, . . . , ?n be its Laplacian eigenvalues. In some recent works a quantity called Laplacian Estrada index was considered, defined as LEE(G)?n1 e?i. We now establish some further properties of LEE, mainly upper and lower bounds in terms of the number of vertices, number of edges, and the first Zagreb index.

scholarly journals Sharp bounds on the signless Laplacian Estrada index of graphs

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 1983-1988 ◽  
Author(s):  
Shan Gao ◽  
Huiqing Liu
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Estrada Index ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The First Zagreb Index

Let G be a connected graph with n vertices and m edges. Let q1, q2,..., qn be the eigenvalues of the signless Laplacian matrix of G, where q1 ? q2 ? ... ? qn. The signless Laplacian Estrada index of G is defined as SLEE(G) = nPi=1 eqi. In this paper, we present some sharp lower bounds for SLEE(G) in terms of the k-degree and the first Zagreb index, respectively.

scholarly journals Lower Bounds for Gaussian Estrada Index of Graphs

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 325 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Magnetic Properties ◽  
Quantum Mechanics ◽  
Lower Bounds ◽  
Graph Spectrum ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Graph Theoretic ◽  
Estrada Index ◽  
Molecular Stability ◽  
The First Zagreb Index

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by λ1,λ2,⋯,λn. The Gaussian Estrada index, denoted by H(G) (Estrada et al., Chaos 27(2017) 023109), can be defined as H(G)=∑i=1ne-λi2. Gaussian Estrada index underlines the eigenvalues close to zero, which plays an important role in chemistry reactions, such as molecular stability and molecular magnetic properties. In a network of particles governed by quantum mechanics, this graph-theoretic index is known to account for the information encoded in the eigenvalues of the Hamiltonian near zero by folding the graph spectrum. In this paper, we establish some new lower bounds for H(G) in terms of the number of vertices, the number of edges, as well as the first Zagreb index.

scholarly journals On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

2016 ◽  
Vol 14 (1) ◽  
pp. 641-648 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Spanning Trees ◽  
Line Graph ◽  
Upper And Lower Bounds ◽  
Barycentric Subdivision ◽  
Electrical Networks ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Vertex Degrees ◽  
Number Of Spanning Trees

Abstract As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).

scholarly journals Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 3031-3042 ◽  
Author(s):  
Ivan Gutman ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

On the distance signless Laplacian Estrada index of graphs

2021 ◽  
pp. 2250078
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Harishchandra Ramane ◽  
Xueliang Li
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Graph Invariants ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
Laplacian Energy ◽  
Signless Laplacian

The distance signless Laplacian eigenvalues [Formula: see text] of a connected graph [Formula: see text] are the eigenvalues 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 define and investigate the distance signless Laplacian Estrada index of a graph [Formula: see text] as [Formula: see text], and obtain some upper and lower bounds for [Formula: see text] in terms of other graph invariants. We also obtain some relations between [Formula: see text] and the auxiliary distance signless Laplacian energy of [Formula: see text].

scholarly journals Energy of Nonsingular Graphs: Improving Lower Bounds

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Hajar Shooshtari ◽  
Jonnathan Rodriguez ◽  
Akbar Jahanbani ◽  
Abbas Shokri
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Adjacency Matrix ◽  
Degree Sequence ◽  
Simple Graph ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
The First Zagreb Index

Let G be a simple graph of order n and A be its adjacency matrix. Let λ 1 ≥ λ 2 ≥ … ≥ λ n be eigenvalues of matrix A . Then, the energy of a graph G is defined as ε G = ∑ i = 1 n λ i . In this paper, we will discuss the new lower bounds for the energy of nonsingular graphs in terms of degree sequence, 2-sequence, the first Zagreb index, and chromatic number. Moreover, we improve some previous well-known bounds for connected nonsingular graphs.

On Energy and Laplacian Energy of Graphs

2016 ◽  
Vol 31 ◽  
pp. 167-186 ◽  
Author(s):  
Kinkar Das ◽  
Seyed Ahmad Mojalal
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
The Absolute ◽  
The First Zagreb Index

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.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

On the first Zagreb Index of Polygraphs

2014 ◽  
Vol 45 ◽  
pp. 147-151 ◽  
Author(s):  
Hossein Shabani ◽  
Reza Kahkeshani
Keyword(s):  
First Zagreb Index ◽  
Zagreb Index ◽  
The First Zagreb Index
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