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.

scholarly journals On the first Zagreb index and multiplicative Zagreb coindices of graphs

2016 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Kinkar Ch. Das ◽  
Nihat Akgunes ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
...  
Keyword(s):  
Degree Sequence ◽  
Molecular Graph ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Irregularity Index ◽  
Vertex Set ◽  
The First Zagreb Index

AbstractFor a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariantsandnamed as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) =. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

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.

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.

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

scholarly journals A Novel/Old Modification of the First Zagreb Index

2018 ◽  
Vol 37 (6-7) ◽  
pp. 1800008 ◽  
Author(s):  
Akbar Ali ◽  
Nenad Trinajstić
Keyword(s):  
First Zagreb Index ◽  
Zagreb Index ◽  
The First Zagreb Index

scholarly journals Bounds on General Randić Index for F-Sum Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xu Li ◽  
Maqsood Ahmad ◽  
Muhammad Javaid ◽  
Muhammad Saeed ◽  
Jia-Bao Liu
Keyword(s):  
Molecular Graph ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Randic Index ◽  
Structure Activity ◽  
General Randić Index ◽  
General Randic Index ◽  
The First Zagreb Index

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

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