Shivaramakrishna Pragada

The switching conjecture for main eigenvalues is asymptotically true

Saieed Akbari, Hitesh Kumar, Bojan Mohar +1

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 (…

A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem

Abhay Jayarajan, M. Rajesh Kannan, Shivaramakrishna Pragada +1

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $λ_{1} (G) \geq \dots \geq λ_{n} (G)$ . Let $c (v)$ denote the maximum order of a clique containing vertex $v$ . We prove the…

Resolution of a problem of Mohar on non-positive inertia

Clive Elphick, Hitesh Kumar, Shivaramakrishna Pragada +1

For a graph $G$ of order $n$ , its positive, negative and non-positive inertia is the number of positive, negative and non-positive eigenvalues of its adjacency matrix $A (G)$ , respe…

math.FA2026

Stability of maximal relative projection constants

Hitesh Kumar, Bojan Mohar, Seyed Ahmad Mojallal +1

For positive integers $n \geq r$ , let $λ (r, n)$ denote the \emph{maximal relative projection constant} of $r$ -dimensional subspaces of $ℓ_{\infty}^{n}$ and $λ (r)$ denote the \emph{maxi…

Localization of the Caro-Wei bound and its applications to bipartiteness

Aida Abiad, Hitesh Kumar, Shivaramakrishna Pragada

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 classi…

Extremal graphs for the $k$ -th eigenvalue

Hitesh Kumar, Bojan Mohar, Seyed Ahmad Mojallal +1

For a simple graph $G$ of order $n$ , let $λ_{1} (G) \geq \dots \geq λ_{n} (G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $λ_{k} (G)$ . A recent theo…

Energy and independence number

Hitesh Kumar, Shivaramakrishna Pragada

For a graph $G$ of order $n$ , with adjacency eigenvalues $λ_{1} (G) \geq \dots \geq λ_{n} (G)$ , the \emph{energy} of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |λ_i(G)|.\] A w…

An improved bound for the strong clique index of graphs

Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada

For a graph $G$ with line graph $L (G)$ , $χ (L (G)^{2})$ and $ω (L (G)^{2})$ are called the \emph{strong chromatic index} and \emph{strong clique index} of $G$ , respectively. A well-known c…

Spectral Radius, Vertex Deletion, and Chromatic Number of Signed Graphs

Abhay Jayarajan, M. Rajesh Kannan, Priti Prasanna Mondal +1

A signed graph $Σ = (G, σ)$ is a graph $G$ with edges given signs $1$ or $- 1$ defined by the function $σ$ . The adjacency matrix of $Σ$ is defined as per these signs. The relation betw…

Localized Turán-type inequalities for $Q$ -index

M. Rajesh Kannan, Hitesh Kumar, Shivaramakrishna Pragada

For a connected graph $G$, let $q (G)$ denote the $Q$ -index of $G$ , i.e., the largest eigenvalue of its signless Laplacian matrix. Abreu and Nikiforov (2013) showed that \[ q(G) \…

Maximum spectral sum of graphs

Hitesh Kumar, Lele Liu, Hermie Monterde +2

For a graph $G$ of order $n$ , the spectral sum of $G$ is defined to be the sum $λ_{1} (G) + λ_{2} (G)$ , where $λ_{1} (G)$ (resp. $λ_{2} (G)$ ) is the largest (resp. second largest) adjacency ei…

Convex combination of first and second eigenvalues of trees

Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada +1

For a graph $G$ , let $λ_{1} (G)$ and $λ_{2} (G)$ denote the largest and the second largest adjacency eigenvalue of $G$ . The sum $λ_{1} (G) + λ_{2} (G)$ is called the \emph{spectral sum} of $G$ …

Localization of spectral Turán-type theorems

M. Rajesh Kannan, Hitesh Kumar, Shivaramakrishna Pragada

Let $G$ be a graph, and let $v$ and $e$ be a vertex and an edge of $G$ , respectively. Define $c (v)$ (resp. $c (e)$ ) to be the order of the largest clique in $G$ containing $v$ (resp…

Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

Alexander Clow, Hitesh Kumar, Shivaramakrishna Pragada

The ultimate independence ratio of a graph $G$ is defined as $I (G) = lim_{k \to \infty} \frac{α ( G ^{□ k} )}{∣ V ( G ) ∣ ^{k}},$ where $α (G^{□ k})$ is the independence…

A new conjecture on the inertia of graphs

Saieed Akbari, Clive Elphick, Hitesh Kumar +2

Let $G$ be a graph with adjacency matrix $A (G)$ . We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^{+} (G)$ and $n^{-} (G)$ denote the number of positive and negative eigen…

Refinement of a conjecture on positive square energy of graphs

Saieed Akbari, Hitesh Kumar, Bojan Mohar +2

Let $G$ be a simple graph of order $n$ with eigenvalues $λ_{1} (G) \geq \dots \geq λ_{n} (G)$ . Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] It was conjec…

Vertex Partitioning and $p$ -Energy of Graphs

Saieed Akbari, Hitesh Kumar, Bojan Mohar +1

For a Hermitian matrix $A$ of order $n$ with eigenvalues $λ_{1} (A) \geq \dots \geq λ_{n} (A)$ , define \[ \mathcal{E}_p^+(A)=\sum_{λ_i > 0} λ_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{λ_i<0} |…

A Linear Lower Bound for the Square Energy of Graphs

Saieed Akbari, Hitesh Kumar, Bojan Mohar +1

Let $G$ be a graph of order $n$ with eigenvalues $λ_{1} \geq \dots \geq λ_{n}$ . Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] The smaller value, $s(G)=\min\{s^+(…

On the second largest adjacency eigenvalue of trees with given diameter

Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada +1

For a graph $G$ , let $λ_{2} (G)$ denote the second largest eigenvalue of the adjacency matrix of $G$ . We determine the extremal trees with maximum/minimum adjacency eigenvalue $λ_{2}$ i…

Bollobás-Nikiforov Conjecture for graphs with not so many triangles

Hitesh Kumar, Shivaramakrishna Pragada

Bollobás and Nikiforov conjectured that for any graph $G \neq = K_{n}$ with $m$ edges \[ λ_1^2+λ_2^2\le \bigg( 1-\frac{1}{ω(G)}\bigg)2m\] where $λ_{1}$ and $λ_{2}$ denote the two largest e…