scholarly journals Some properties of the Zagreb indices

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2667-2675
Author(s):  
Emina Milovanovic ◽  
Igor Milovanovic ◽  
Muhammad Jamil
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Graph ◽  
Graph Invariants ◽  
Zagreb Indices

Let G = (V,E), V = {1,2,..., n}, E = {e1,e2,..., em}, be a simple graph with n vertices and m edges. Denote by d1 ? d2 ? ... ? dn > 0, and d(e1) ? d(e2) ? d(em), sequences of vertex and edge degrees, respectively. If i-th and j-th vertices of G are adjacent, it is denoted as i ~ j. Graph invariants referred to as the first, second and the first reformulated Zagreb indices are defined as M1=?ni=1 di2, M2= ?i~j didj and EM1 = ?mi=1 d(ei)2, respectively. Let ?1 ? ?2? ... ?n be eigenvalues of G. With ?(G) = ?1 a spectral radius of G is denoted. Lower bounds for invariants M1, M2, EM1 and ?(G) are obtained.

A note on the relationship between graph energy and determinant of adjacency matrix

2019 ◽  
Vol 11 (01) ◽  
pp. 1950001
Author(s):  
Igor Ž. Milovanović ◽  
Emina I. Milovanović ◽  
Marjan M. Matejić ◽  
Akbar Ali
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Graph Invariants ◽  
Graph Energy ◽  
The Matrix ◽  
Matrix Formula ◽  
The Relationship

Let [Formula: see text] be a simple graph of order [Formula: see text], without isolated vertices. Denote by [Formula: see text] the adjacency matrix of [Formula: see text]. Eigenvalues of the matrix [Formula: see text], [Formula: see text], form the spectrum of the graph [Formula: see text]. An important spectrum-based invariant is the graph energy, defined as [Formula: see text]. The determinant of the matrix [Formula: see text] can be calculated as [Formula: see text]. Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem. 77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants [Formula: see text] and [Formula: see text] are derived. Consequently, all the bounds established in the aforementioned paper are improved.

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

2019 ◽  
Vol 12 (01) ◽  
pp. 2050006 ◽  
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
E. Hashemi ◽  
S. Paul
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Graph Invariants ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Eigenvalues Of Graphs

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.

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.

scholarly journals Some lower bounds for the spectral radius of matrices using traces

2010 ◽  
Vol 432 (4) ◽  
pp. 1007-1016 ◽  
Author(s):  
Lin Wang ◽  
Mao-Zhi Xu ◽  
Ting-Zhu Huang
Keyword(s):  
Spectral Radius ◽  
Lower Bounds

scholarly journals Cauchy's interlace theorem and lower bounds for the spectral radius

2000 ◽  
Vol 23 (8) ◽  
pp. 563-566 ◽  
Author(s):  
A. McD. Mercer ◽  
Peter R. Mercer
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Proof ◽  
Symmetric Matrix ◽  
Real Symmetric Matrix

We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

Lower bounds for the Laplacian spectral radius of graphs

2021 ◽  
Vol 631 ◽  
pp. 136-142
Author(s):  
B. Afshari ◽  
M.T. Saadati ◽  
R. Saadati
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Laplacian Spectral Radius
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