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.

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 Lower bound for the spectral radius of the starlike trees

2021 ◽  
Vol 2090 (1) ◽  
pp. 012127
Author(s):  
Rubí Arrizaga-Zercovich
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Largest Eigenvalue ◽  
Acyclic Graph ◽  
Starlike Tree ◽  
The Largest Eigenvalue

Abstract A tree is a connected acyclic graph. A tree is called a starlike if exactly one of its vertices has degree greater than two. Let λι be the largest eigenvalue of the adjacency matrix of a starlike tree. In this work, we obtain a lower bound for the spectral radius of a starlike tree. This bound only depends of the maximum degree of the vertices.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

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).

scholarly journals Regular Factors of Regular Graphs from Eigenvalues

10.37236/431 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hongliang Lu
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Third ◽  
K Factor ◽  
Second Largest Eigenvalue

Let $r$ and $m$ be two integers such that $r\geq m$. Let $H$ be a graph with order $|H|$, size $e$ and maximum degree $r$ such that $2e\geq |H|r-m$. We find a best lower bound on spectral radius of graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular graph of order $|G|$ and $ k < r$ be an integer. Using the previous results, we find some best upper bounds (in terms of $r$ and $k$) on the third largest eigenvalue that is sufficient to guarantee that $G$ has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that $G$ is $k$-critical when $k|G|$ is odd. Our results extend the work of Cioabă, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.

scholarly journals A note on a walk-based inequality for the index of a signed graph

2021 ◽  
Vol 9 (1) ◽  
pp. 19-21
Author(s):  
Zoran Stanić
Keyword(s):  
Adjacency Matrix ◽  
Signed Graph ◽  
Arbitrary Length ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

scholarly journals A note on distance spectral radius of trees

2017 ◽  
Vol 5 (1) ◽  
pp. 296-300
Author(s):  
Yanna Wang ◽  
Rundan Xing ◽  
Bo Zhou ◽  
Fengming Dong
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Largest Eigenvalue ◽  
Maximal Distance ◽  
The Largest Eigenvalue

Abstract The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

scholarly journals The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen Li ◽  
Michael K. Ng
Keyword(s):  
Spectral Radius ◽  
Perturbation Analysis ◽  
Absolute Difference ◽  
Nonnegative Tensor ◽  
Backward Error ◽  
Perturbation Bound ◽  
Error Matrix ◽  
Largest Eigenvalue ◽  
The Stability ◽  
The Largest Eigenvalue

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor A. The main contribution of this paper is to show that when A is perturbed to a nonnegative tensor A~ by ΔA, the absolute difference between the spectral radii of A and A~ is bounded by the largest magnitude of the ratio of the ith component of ΔAxm-1 and the ith component xm-1, where x is an eigenvector associated with the largest eigenvalue of A in magnitude and its entries are positive. We further derive the bound in terms of the entries of A only when x is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix ΔA and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

Some Inequalities for the Largest Eigenvalue of a Graph

2002 ◽  
Vol 11 (2) ◽  
pp. 179-189 ◽  
Author(s):  
V. NIKIFOROV
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Minimal Degree ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free thenThis inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We showLet δ be the minimal degree of G. We showThis inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

scholarly journals Upper bounds on Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

2017 ◽  
Vol 32 ◽  
pp. 447-453
Author(s):  
Qi Kong ◽  
Ligong Wang
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Strongly Regular Graph ◽  
Upper Bounds ◽  
Free Graph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Strongly Regular

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. The results are the following: (1) Let $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

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