scholarly journals On the spectral invariants of symmetric matrices with applications in the spectral graph theory

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2925-2932
Author(s):  
Abdullah Alazemi ◽  
Milica Andjelic ◽  
Slobodan Simic
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Weighted Graph ◽  
Spectral Graph Theory ◽  
Combinatorial Structure ◽  
Symmetric Matrices ◽  
Spectral Invariants ◽  
Spectral Graph ◽  
Principal Submatrices ◽  
The Relationship

We first prove a formula which relates the characteristic polynomial of a matrix (or of a weighted graph), and some invariants obtained from its principal submatrices (resp. vertex deleted subgraphs). Consequently, we express the spectral radius of the observed objects in the form of power series. In particular, as is relevant for the spectral graph theory, we reveal the relationship between spectral radius of a simple graph and its combinatorial structure by counting certain walks in any of its vertex deleted subgraphs. Some computational results are also included in the paper.

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.

scholarly journals The Kirchhoff Index of Hypercubes and Related Complex Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jiabao Liu ◽  
Jinde Cao ◽  
Xiang-Feng Pan ◽  
Ahmed Elaiw
Keyword(s):  
Graph Theory ◽  
Complex Networks ◽  
Laplacian Matrix ◽  
Exact Formula ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Relationship Of ◽  
The Relationship

The resistance distance between any two vertices ofGis defined as the network effective resistance between them if each edge ofGis replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices inG. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networksQnby utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networksQnand its three variant networksl(Qn),s(Qn),t(Qn)by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes ofl(Qn),s(Qn), andt(Qn)were proposed, respectively.

Sign in / Sign up

Export Citation Format

Share Document