scholarly journals Degenerate Turán Problems for Hereditary Properties

10.37236/6775 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Vladimir Nikiforov ◽  
Michael Tait ◽  
Craig Timmons
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Vertex Graph ◽  
Turan Problems ◽  
Hereditary Properties

Let $H$ be a graph and $t\geqslant s\geqslant 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $\lambda(G) = O\left(n^{1-1/s}\right)$ where $\lambda(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.

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 New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

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.

Trees with given maximum degree minimizing the spectral radius

2016 ◽  
Vol 31 ◽  
pp. 335-361
Author(s):  
Xue Du ◽  
Lingsheng Shi
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Equal Length ◽  
Disjoint Paths ◽  
Common Vertex ◽  
Largest Eigenvalue ◽  
Large N ◽  
The Largest Eigenvalue

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

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 The Spectral Radius and the Maximum Degree of Irregular Graphs

10.37236/956 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Sebastian M. Cioabă
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Largest Eigenvalue ◽  
Irregular Graphs ◽  
The Largest Eigenvalue

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1>{1\over nD}, $$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

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 $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{1\over nD}. $$

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