arXiv subjects

Vladimir Nikiforov

Publications and source records attributed to Vladimir Nikiforov.

At least 19 recordsLinked to original sources

arXiv2022

On graphs with eigenvectors in $\{1, -1, 0\}$ and the max $k$-cut problem

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max $k$-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite families of extremal graphs for the obtained upper bounds.

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arXiv2021

The energy of C4-free graphs of bounded degree

Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph is the sum of the moduli of its eigenvalues. We give more general theorems and state two conjectures.

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arXiv2017

On the $α$-index of graphs with pendent paths

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right] $, write $A_α\left( G\right) $ for the matrix \[ A_α\left( G\right) =αD\left( G\right) +(1-α)A\left( G\right) . \] This paper presents some extremal results about the spectral radius $ρ_α\left( G\right) $ of $A_α\left( G\right) $ that generalize previous results about $ρ_{0}\left( G\right) $ and $ρ_{1/2}\left( G\right) $. In particular, write $B_{p,q,r}$ be the graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. It is shown that if $α\in\left[ 0,1\right) $ and $G$ is a graph of order $n$ and diameter at least $k,$ then% \[ ρ_α(G)\leqρ_α(B_{n-k+2,\lfloor k/2\rfloor,\lceil k/2\rceil}), \] with equality holding if and only if $G=B_{n-k+2,\lfloor k/2\rfloor,\lceil k/2\rceil}$. This result generalizes results of Hansen and Stevanović \cite{HaSt08}, and Liu and Lu \cite{LiLu14}.

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arXiv2017

On the minimum trace norm of (0,1)-matrices

The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $ψ_{n}\left( m\right) $ of $\left( 0,1\right) $-matrices of size $n\times n$ with exactly $m$ ones. It is shown that: (1) if $n\geq2$ and $n<m\leq2n,$ then $ψ_{n}\left( m\right) \leq \sqrt{m+\sqrt{2\left( m-1\right) }}$ , with equality if and only if $m$ is a prime; (2) if $n\geq4$ and $2n<m\leq3n,$ then $ψ_{n}\left( m\right) \leq \sqrt{m+2\sqrt{2\left\lfloor m/3\right\rfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime; (3) if $3n<m\leq4n,$ then $ψ_{n}\left( m\right) \leq\sqrt{m+2\sqrt{m-2}}% $ , with equality if and only if there is an integer $k\geq1$ such that $m=12k\pm2$ and $4k\pm1,6k\pm1,12k\pm1$ are primes.

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arXiv2017

More spectral bounds on the clique and independence numbers

We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues.

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arXiv2017

Degenerate Turán problems for hereditary properties

Let $H$ be a graph and $t\geq s\geq 2$ 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 $λ(G) = O\left(n^{1-1/s}\right)$ where $λ(G)$ 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 $K_{s,t}$.

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arXiv2016

Spectral radius and Hamiltonicity of graphs with large minimum degree

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $λ\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $δ\left( G\right) \geq k.$ If \[ λ\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $δ\left( G\right) \geq k.$ If \[ λ\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless $G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.

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arXiv2016

A note on the positive semidefinitness of $A_α(G)$

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right] $, write $A_α\left( G\right) $ for the matrix \[ A_α\left( G\right) =αD\left( G\right) +(1-α)A\left( G\right) . \] Let $α_{0}\left( G\right) $ be the smallest $α$ for which $A_α(G)$ is positive semidefinite. It is known that $α_{0}\left( G\right) \leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then \[ α_{0}=\frac{-λ_{\min}(A(G))}{d-λ_{\min}(A(G))}, \] where $λ_{\min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contains a bipartite component if and only if $α_{0}\left( G\right) =1/2$; (3) if $G$ is $r$-colorable, then $α_{0}\left( G\right) \geq1/r$.

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arXiv2016

On the $A_α$-spectra of trees

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right],$ define the matrix $A_α\left(G\right) $ as \[ A_α\left(G\right) =αD\left(G\right) +(1-α)A\left(G\right) \] where $0\leqα\leq1$. This paper gives several results about the $A_α$-matrices of trees. In particular, it is shown that if $T_Δ$ is a tree of maximal degree $Δ,$ then the spectral radius of $A_α(T_Δ)$ satisfies the tight inequality \[ ρ(A_α(T_Δ))<αΔ+2(1-α)\sqrt{Δ-1}. \] This bound extends previous bounds of Godsil, Lovász, and Stevanović. The proof is based on some new results about the $A_α$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_α$ of general graphs are proved, implying tight bounds for paths and Bethe trees.

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arXiv2016

Beyond graph energy: norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.

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arXiv2015

The clique number and the smallest Q-eigenvalue of graphs

Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order $n,$ with no complete subgraph of order $r+1?$ It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on $q_{\min}$ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum $q_{\min}$ is established. As a supporting tool, the spectra of the Laplacian and the signless Laplacian of blowups of graphs are calculated.

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arXiv2015

Maxima of the Q-index: graphs with no K_s,t

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order $n$ that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases. More precisely, it is shown that if $G$ is a graph of order $n,$ with no subgraph isomorphic to $K_{2,s+1},$ then the largest eigenvalue $q(G)$ of the signless Laplacian of $G$ satisfies \[ q(G)\leq\frac{n+2s}{2}+\frac{1}{2}\sqrt{(n-2s)^{2}+8s}, \] with equality holding if and only if $G$ is a join of $K_{1}$ and an $s$-regular graph of order $n-1.$

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arXiv2015

Extrema of graph eigenvalues

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $λ_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2 2,$ and even tighter bounds are obtained for the $k$'th largest singular value $λ_{k}^{\ast}(G).$ Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can $λ_{k}(G)+λ_{k}(\bar{G})$ be$?$ These constructions are successful also in another open question: How large can the Ky Fan norm $λ_{1}^{\ast}(G)+...+λ_{k}^{\ast }(G)$ be $?$ Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for $(-1,1)$-matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study.

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arXiv2014

Graph functions maximized on a path

Given a connected graph $G\ $of order $n$ and a nonnegative symmetric matrix $A=\left[ a_{i,j}\right] $ of order $n,$ define the function $F_{A}\left( G\right) $ as% \[ F_{A}\left( G\right) =\sum_{1\leq i<j\leq n}d_{G}\left( i,j\right) a_{i,j}, \] where $d_{G}\left( i,j\right) $ denotes the distance between the vertices $i$ and $j$ in $G.$ In this note it is shown that $F_{A}\left( G\right) \leq F_{A}\left( P\right) \,$for some path of order $n.$ Moreover, if each row of $A$ has at most one zero off-diagonal entry, then $F_{A}\left( G\right) <F_{A}\left( P\right) \,$for some path of order $n,$ unless $G$ itself is a path. In particular, this result implies two conjectures of Aouchiche and Hansen: - the spectral radius of the distance Laplacian of a connected graph $G$ of order $n$ is maximal if and only if $G$ is a path; - the spectral radius of the distance signless Laplacian of a connected graph $G$ of order $n$ is maximal if and only if $G$ is a path.

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arXiv2014

Spectra of the blow-up graphs

Let $G$ be graph on $n$ vertices and $G^{(t)}$ its blow-up graph of order $t.$ In this paper, we determine all eigenvalues of the Laplacian and the signless Laplacian matrix of $G^{(t)}$ and its complement $\bar{G^{(t)}}.$

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arXiv2014

Maxima of the Q-index: forbidden even cycles

Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be the largest eigenvalue of the signless Laplacian of $G$. Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order $n-k;$ and let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}.$ It is shown that if $k\geq2,$ $n\geq400k^{2},$ and $G$ is a graph of order $n,$ with no cycle of length $2k+2,$ then $q\left( G\right) <q\left( S_{n,k}^{+}\right) ,$ unless $G=S_{n,k}^{+}.$ This result completes the proof of a conjecture of de Freitas, Nikiforov and Patuzzi.

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arXiv2014

Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ λ^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} . \] The most remarkable feature of $λ^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $λ^{\left( 1\right) }$ is the Lagrangian, $λ^{\left( 2\right) }$ is the spectral radius and $λ^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $λ^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right) $ be the $r$-partite Turán graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then \[ λ^{\left( p\right) }\left( G\right) <λ^{\left( p\right) }\left( T_{r}\left( n\right) \right) , \] unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\ $is a graph of order $n,$ with \[ λ^{\left( p\right) }\left( G\right) >λ^{\left( p\right) }\left( T_{r}\left( n\right) \right) , \] then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$

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