Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $λ_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for $D_{\max}(G)-λ_1(G)$, and characterize the extremal graphs attaining the bound. As a corollary, we solve a conjecture posed by Liu, Shu and Xue.
A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a conjecture on the distance spectral radius of cacti in [S.S. Bose, M. Nath, S. Paul, On the distance spectral radius of cacti, Linear Algebra Appl. 437 (2012) 2128--2141]. We prove the result in the context of hypertrees.
We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Turán type result which might turn out to be of independent interest.
Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of $G^σ$ is the same for every orientation $σ$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. This paper, by using the Kelmans transformation, …
A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique grap…
The Randić index $R(G)$ of a graph $G$ is defined as the sum of $(d_i d_j)^{-1/2}$ over all edges $v_i v_j$ of $G$, where $d_i$ is the degree of the vertex $v_i$ in $G$. The radius $r(G)$ of a graph $G$ is the minimum graph eccentricity of any graph vertex in $G$. Fajtlowicz(1988) conjectures $R(G) \ge r(G)-1$ for all connected graph $G$. A stronger version, $R(G) \ge r(G)$, is conjectured by Caporossi and Hansen(2000) for all connected graphs except even paths. In this paper, we make use of Harmonic index $H(G)$, which is defined as the sum of $\frac{2}{d_i+d_j}$ over all edges $v_i v_j$ of $…
This note proves that, given one member, $T$, of a particular family of radius-three trees, every radius-two, triangle-free graph, $G$, with large enough chromatic number contains an induced copy of $T$.
In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].
Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdière parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph w…
Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four.
Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a …
For a connected graph $G$, the distance Laplacian spectral radius of $G$ is the spectral radius of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. In this paper, we determine the unique graphs with maximum distance Laplacian spectral radius among unicyclic graphs.
The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjectu…
Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $Δ$. Let $λ_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the largest eigenvalue of non-regular graphs, J. Combin. Theory Ser. B, 97 (2007) 1010--1018] asserts that \[ \lim_{n\to\infty} \frac{n^2(Δ-λ_1(G))}{Δ-1} = π^2 \] for each fixed $Δ$. Concerning an important structural property of the extremal graphs $G$, Liu and Li present another conjecture which states that $G$ has degree sequence $Δ,\ldots,Δ,δ$. Here, $δ=Δ-1$ or $δ=Δ-2$…
Let $\mathcal{S}_n$ denote the set of permutations of $\{1,2,\dots,n\}$. The function $f(n,s)$ is defined to be the minimum size of a subset $S\subseteq \mathcal{S}_n$ with the property that for any $ρ\in \mathcal{S}_n$ there exists some $σ\in S$ such that the Hamming distance between $ρ$ and $σ$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kézdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kézdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!<…
In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We also state an analog of the conjecture of Chen et al for digraphs and prove it for sufficiently large order.
For a connected graph $G$ and $α\in [0,1)$, the distance $α$-spectral radius of $G$ is the spectral radius of the matrix $D_α(G)$ defined as $D_α(G)=αT(G)+(1-α)D(G)$, where $T(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. We give bounds for the distance $α$-spectral radius, especially for graphs that are not transmission regular, propose some graft transformations that decrease or increase the distance $α$-spectral radius, and determine the unique graphs with minimum and maximum distance $α$-spectral radius among some classes of graphs.
The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance dis…
Let $μ(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. Let $S_{n,k}=K_{k}\vee\overline{K_{n-k}}$, and let $S_{n,k}^{+}$ be the graph obtained from $S_{n,k}$ by adding a single edge joining two vertices of the independent set of $S_{n,k}$. In 2010, Nikiforov conjectured that for a given integer $k$, every graph $G$ of sufficiently large order $n$ with $μ(G)\geq μ(S_{n,k}^{+})$ contains …
Write $ρ\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $ρ\left( G\right) $ for $r=2$, and found an asymptotically tight bound on $ρ\left( G\right) $ for $r\geq3.$ They also conjectured that \[ ρ\left( G\right) >ρ\left( S_{n,r-1}\right) , \] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.
We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of $P_2$ and $P_{n-2}$ is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and $P_{n-1}$. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineappl…
The goal of this paper is twofold; first, show the equivalence between certain problems in geometry, such as view-obstruction and billiard ball motions, with the estimation of covering radii of lattice zonotopes. Second, we will estimate upper bounds of said radii by virtue of the Flatness Theorem. These problems are similar in nature with the famous lonely runner conjecture.
Given a connected graph $G$, the Randić index $R(G)$ is the sum of $\tfrac{1}{\sqrt{d(u)d(v)}}$ over all edges $\{u,v\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest eigenvalue of the singless Laplacian matrix of $G$ and $n=|V(G)|$. Hansen and Lucas (2010) made the following conjecture: \[ \frac{q(G)}{R(G)} \leq \begin{cases} \frac{4n-4}{n} & 4 \leq n\leq 12 \frac{n}{\sqrt{n-1}} & n\geq 13 \end{cases} \] with equality if and only if $G=K_{n}$ for $4\leq n\leq 12$ and $G=S_n$ for $n\geq 13$, respectively. Deng, Balachandran, and …
For a simple graph $F$, let $\mathrm{Ex}(n, F)$ and $\mathrm{Ex_{sp}}(n,F)$ denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. The Turán graph $T_{n,r}$ is the complete $r$-partite graph on $n$ vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let $F$ be any graph such that the graphs in $\mathrm{Ex}(n,F)$ are Tur…
We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino \& Schymura (2017) that the $d$-th covering minimum of the standard terminal $n$-simplex equals $d/2$, for every $n>d$. We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the cover…
The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_α(G)=αTr(G)+(1-α)D(G), \;\;0\leα\le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs w…
In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for some } h\in E(\mathcal{H})\}$. A graph is \textit{outerplanar} if it can be embedded in the plane such that all its vertices lie on the outer face. A $3$-uniform hypergraph $\mathcal{H}$ is called \textit{outerplanar} if its shadow has an outerplanar embedding such that every hyperedge of $\mathcal{H}$ is the vertex set of an interior triangular face of the sha…
In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We show the connection between minimizing the total distance and maximizing the size under the same conditions. We also prove some asymptotically optimal bounds for the maximum total distance.
Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ (or $μ(G)\ge μ(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$, where $C_\ell$ is a cycle of length $\ell$ and $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join graph of a complete graph of order $k$ and an empty graph on $n-k$ vertices, and $S_{n,k}^+$ is the graph obtained from $S_{n,k}$ by adding an edge in the independent set of $S_{n,k}$. %This can be vie as spectral version of Erdös and Sós conj…
The spectral radius $ρ(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \sim 1 \sim 2 \sim ... \sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\in\{1,2,...,t\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioabǎ-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed $e\…
It is well known that $n/(n - μ)$, where $μ$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $μ$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the sum of the squares of the positive eigenvalues. We prove this conjecture for various classes of graphs, including triangle-free graphs, and for almost all graphs.
The Brauldi-Hoffman conjecture, proved by Rowlinson in 1988, characterized the graph with maximal spectral radius among all simple graphs with prescribed number of edges. In 2008, Bhattacharya, Friedland, and Peled proposed an analog, which will be called the BFP conjecture in the following, of the Brauldi-Hoffman conjecture for the bipartite graphs with fixed numbers of edges in the graph and vertices in the bipartition. The BFP conjecture was proved to be correct if the number of edges is large enough by several authors. However, in this paper we provide some counterexamples of the BFP conje…
Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erdős-Sós conjecture.
Nonregular graphs of odd maximum degree with maximum spectral radius
School of Mathematics, Nanjing University（南京大学数学系）；Department of Mathematics, Shanghai University（上海大学数学系）；Newtouch Center for Mathematics of Shanghai University（上海大学新触数学中心）
We study the Randic index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We study radius and diameter for cacti then verify the radius bound and strengthen two diameter bounds for cacti. Along the way, we produce several other bounds for the Randic index in terms of graph size, order, and valency for several special classes of graphs, including chemical nontrivial cacti and cacti with starlike BC-trees.
Given a connected graph $G=(V,E)$ and a vertex set $S\subset V$, the {\em Steiner distance} $d(S)$ of $S$ is the size of a minimum spanning tree of $S$ in $G$. For a connected graph $G$ of order $n$ and an integer $k$ with $2\leq k \leq n$, the $k$-eccentricity of a vertex $v$ in $G$ is the maximum value of $d(S)$ over all $S\subset V$ with $|S|=k$ and $v\in S$. The minimum $k$-eccentricity, ${srad}_k(G)$, is called the $k$-radius of $G$ while the maximum $k$-eccentricity, ${sdiam}_k(G)$, is called the $k$-diameter of $G$. In 1990, Henning, Oellermann, and Swart [\textit{Ars Combinatoria} \tex…
Asymptotic maximum spectral radius of nonregular graphs
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092, P.R. China（同济大学）
A well known upper bound for the spectral radius of a graph, due to Hong, is that $μ_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete $q$-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.
For $r\geq 2$ and $p\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$ρ_p(H)=\max_{{\bf x}\in \mathbb{R}^n: \|{\bf x}\|_p=1}r \cdot \!\!\!\! \sum_{\{i_1,i_2,\ldots, i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r},$$ where the maximum is taken over all ${\bf x\in \mathbb{R}^n}$ with the $p$-norm equals 1. In this paper, we proved for any integer $r\geq 2$, and any real $p\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\choose r}$ edges (for some real $s\geq r-1$), we have $$λ_p(H)\leq \frac{rm}{s^{r/p}}.$$ The equality holds if and only…
The Gyárfás-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest $H$, if $G$ is $H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded…
The distance matrix $\mathcal{D}$ of a connected graph $G$ is the matrix indexed by the vertices of $G$ which entry $\mathcal{D}_{i,j}$ equals the distance between the vertices $v_i$ and $v_j$. The distance signless Laplacian matrix $\mathcal{Q}(G)$ of graph $G$ is defined as $\mathcal{Q}(G)=Diag(Tr)+\mathcal{D}(G)$, where $Diag(Tr)$ is the diagonal matrix of the vertex transmissions in $G$. The largest eigenvalue of $\mathcal{Q}(G)$ is called the distance signless Laplacian spectral radius of $G$, written as $η_1(G)$. And a perfect matching in a graph is a set of disadjacent edges covering ev…
In 1990, Cvetković and Rowlinson [The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that among all outerplanar graphs on $n$ vertices, $K_1\vee P_{n-1}$ attains the maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed the conjecture for sufficiently large values of $n$. In this article, we show the conjecture is true for all $n\geq2$ except for $n=6$.
A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension $n\geq 3$ and radius $r\geq 2$. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further…
László Fejes Tóth and Aladár Heppes proposed the following generalization of the kissing number problem. Given a ball in $\mathbb{R}^d$, consider a family of balls touching it, and another family of balls touching the first family. Find the maximal possible number of balls in this arrangement, provided that no two balls intersect by interiors, and all balls are congruent. They showed that the answer for disks on the plane is $19$. They also conjectured that if there are three families of disks instead of two, the answer is $37$. In this paper we confirm this conjecture.
We give a new bound on the probability that the random sum $ξ_1 v_1 +...+ ξ_n v_n$ belongs to a ball of fixed radius, where the $ξ_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove a conjecture of Frankl and Füredi (raised in 1988), which can be seen as the high dimensional version of the classical Littlewood-Offord-Erd\H os theorem.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_α(G)$ as $A_α(G)=αD(G)+(1-α)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\leq i\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$. Let $s=0,1$. Nikiforov and R…
Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance, in other words that there are no outliers in the circular law. In this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses th…
