Using spectral and polynomial methods, we show several sharp spectral bounds for the clique and chromatic numbers of graph powers that only use spectral information from the original graph. In particular, we strengthen earlier spectral-radius estimates to obtain an asymptotically sharp colouring bound for general graph...
A. Abiad, Gabriel Coutinho, Hitesh Kumar et al.· 0 citations
For positive integers $n\ge r$, let $\lambda(r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of $\ell_\infty^n$ and $\lambda(r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $\lambda(r,n)$ is a non-decreasing sequence...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound...
Swaroop G. Hegde, Hitesh Kumar, Pratibha· 0 citations
An eigenvalue of a signed graph is called \emph{main} if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, Fran\c{c}a, Ghasemian, Javarsineh, and de Lima (2021) cons...
S. Akbari, Hitesh Kumar, Bojan Mohar et al.· 1 citation
We confirm a conjecture of Brause, Randerath, Rautenbach and Schiermeyer (2016) by proving a localized lower bound on the independence number of a graph that strengthens the classical bounds of Fajtlowicz (1978) and of Caro (1979) and Wei (1981), which in turn settles a conjecture by Bertram and Hor\'{a}k (1996). Our p...
A. Abiad, Hitesh Kumar, Shivaramakrishna Pragada· 3 citations
For a graph $G$ of order $n$, with adjacency eigenvalues $\lambda_1(G) \geq \cdots \geq \lambda_n(G)$, the \emph{energy} of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |\lambda_i(G)|.\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \[\mathcal{E}(G) \ge 2\left(n-\alpha(G)\r...
For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The \emph{cyclotomic conjecture} proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, G...
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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