scholarly journals Revisiting Two Classical Results on Graph Spectra

10.37236/932 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Graph Spectra ◽  
Proper Subgraph

Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ be the largest and smallest eigenvalues of the adjacency matrix of a graph $G$. Our main results are: (i) If $H$ is a proper subgraph of a connected graph $G$ of order $n$ and diameter $D$, then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over\mu\left( G\right) ^{2D}n}. $$ (ii) If $G$ is a connected nonbipartite graph of order $n$ and diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{2\over\mu\left( G\right) ^{2D}n}. $$ For large $\mu $ and $D$ these bounds are close to the best possible ones.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

scholarly journals Some New Results on Various Graph Energies of the Splitting Graph

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Zheng-Qing Chu ◽  
Saima Nazeer ◽  
Tariq Javed Zia ◽  
Imran Ahmed ◽  
Sana Shahid
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Absolute Value ◽  
Regular Splitting ◽  
The Absolute ◽  
Splitting Graph ◽  
Simple Connected Graph

The energy of a simple connected graph G is equal to the sum of the absolute value of eigenvalues of the graph G where the eigenvalue of a graph G is the eigenvalue of its adjacency matrix AG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graph S′G is a multiple of corresponding energy of a given graph G.

Resilient Design of Complex Engineered Systems

2013 ◽  
Author(s):  
Hoda Mehrpouyan ◽  
Brandon Haley ◽  
Andy Dong ◽  
Irem Y. Tumer ◽  
Chris Hoyle
Keyword(s):  
Complex Network ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Graph Spectra ◽  
Complex Engineered Systems ◽  
Resilient Design ◽  
Spectra Properties ◽  
Key Driver ◽  
Engineered Systems

This paper presents a complex network and graph spectral approach to calculate the resiliency of complex engineered systems. Resiliency is a key driver in how systems are developed to operate in an unexpected operating environment, and how systems change and respond to the environments in which they operate. This paper deduces resiliency properties of complex engineered systems based on graph spectra calculated from their adjacency matrix representations, which describes the physical connections between components in a complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices also have eigenvalue and eigenspectrum properties that can be used to calculate the resiliency of the complex engineered system. One such property of the Laplacian matrix is the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph spectra properties to predict the resiliency of the system under design.

scholarly journals Spectral conditions of existence of the graph circumference

2019 ◽  
Vol 55 (2) ◽  
pp. 169-175
Author(s):  
V. I. Benediktovich
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Arbitrary Graph ◽  
Laplace Matrix ◽  
Graph Eigenvalues

A graph parameter – a circumference of a graph – and its relationship with the algebraic parameters of a graph – eigenvalues of the adjacency matrix and the unsigned Laplace matrix of a graph – are considered in this article. Earlier we have obtained the lower estimates of the spectral radius of an arbitrary graph and a bipartitebalanced graph for existence of the Hamiltonian cycle in it. Recently the problem of existence of a cycle of length n – 1 in a graph depending on the values of its above-mentioned spectral radii has been investigated. This article studies the problem of existence of a cycle of length n – 2 in a graph depending on the lower estimates of the values of its spectral radius and the spectral radius of its unsigned Laplacian and the spectral conditions of existence of the circumference of a graph (2-connected graph) are obtained.

scholarly journals On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2395
Author(s):  
Wenjie Ning ◽  
Kun Wang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Estrada Index

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

Tetracyclic graphs with maximal Estrada index

2017 ◽  
Vol 09 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Nader Jafari Rad ◽  
Akbar Jahanbani ◽  
Doost Ali Mojdeh
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Estrada Index ◽  
Simple Connected Graph

The Estrada index of a simple connected graph [Formula: see text] of order [Formula: see text] is defined as [Formula: see text], where [Formula: see text] are the eigenvalues of the adjacency matrix of [Formula: see text]. In this paper, we characterize all tetracyclic graphs of order [Formula: see text] with maximal Estrada index.

scholarly journals The Spectral Radius of Subgraphs of Regular Graphs

10.37236/1021 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Regular Graphs ◽  
Proper Subgraph

Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ be the largest and smallest eigenvalues of the adjacency matrix of a graph $G$. Our main results are: (i) Let $G$ be a regular graph of order $n$ and finite diameter $D.$ If $H$ is a proper subgraph of $G,$ then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over nD}. $$ (ii) If $G$ is a regular nonbipartite graph of order $n$ and finite diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{1\over nD}. $$

scholarly journals On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

2021 ◽  
Vol 45 (02) ◽  
pp. 299-307
Author(s):  
HANYUAN DENG ◽  
TOMÁŠ VETRÍK ◽  
SELVARAJ BALACHANDRAN
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectral Radius ◽  
Harmonic Index ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
The Matrix ◽  
The Relationship

The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.

scholarly journals On the Spectrum of Threshold Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Irene Sciriha ◽  
Stephanie Farrugia
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Equitable Partition ◽  
Combinatorial Properties ◽  
Threshold Graphs ◽  
Vertex Degrees ◽  
Positive Integers

The antiregular connected graph on r vertices is defined as the connected graph whose vertex degrees take the values of r−1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number r of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.

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