arXiv subjects

M. Rajesh Kannan

Publications and source records attributed to M. Rajesh Kannan.

At least 19 recordsLinked to original sources

arXiv2026

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

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 vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{λ_i(G)>0}λ_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.

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arXiv2026

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

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 between the largest eigenvalue of $G$ and $G-v$ has been studied in recent years, where $G-v$ is the graph obtained from $G$ by deleting the vertex $v$. In 2020, Sun and Das proved that the difference of the squares of the largest eigenvalues of the graphs $G$ and $G-v$ is bounded above by $2d(v)-1$ where $d(v)$ is the degree of $v$. A similar result need not be true for the largest eigenvalue of signed graphs. In this paper, we prove that the result is valid for the spectral radius of signed graphs. On the other hand, the signed graph version of Hoffman's chromatic number bound was proved by Wang et al. in 2021. They also discussed the difficulty in proving the extended version encompassing all eigenvalues of $Σ$ as was done for unsigned graphs by Wocjan and Elphick. We note down a consequence of Wocjan and Elphick's lower bound for the chromatic number in terms of all the eigenvalues of $Σ_+$; all the eigenvalues of $Σ$ and $Σ_-$, where $Σ_+$ (resp. $Σ_-$) is the spanning subgraph induced by the positive (resp. negative) edges. We give examples where the result fails even under various restrictions on the signed graph. Finally, we improve an upper bound for the $k$-th power of the largest eigenvalue given by Stanić in terms of walks in signed graphs and give lower bounds for the least eigenvalue in terms of various parameters of $Σ_+$ and $Σ_-$.

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arXiv2026

Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov

The algebraic connectivity of a graph $G$ is a well-studied graph invariant that is related to other properties of the graph such as connectivity and expansion. Given $n$ and $m$, $α(n,m)$ is the maximum algebraic connectivity of a graph with $n$ vertices and $m$ edges. In 2015, Kolokolnikov conjectured that $α(n,2n-4)=2$ for $n\geq 4$, and verified this claim computationally for $n \le 12$. In this paper, we prove Kolokolnikov's conjecture. We also show that $α(n,3(n-3)) = 3$ is false in general. %Combined with the computational verification for $n \le 12$, this yields $α(n,2n-4)=2$ for all admissible values of $n$.

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arXiv2026

On the $l_\infty$-analog of Algebraic Connectivity

The algebraic connectivity of a graph, defined as the second smallest eigenvalue of its Laplacian matrix, admits a well-known variational characterization involving the $\ell_2$-norm. Motivated by the recent introduction of its $\ell_\infty$-analogue by Andrade and Dahl, we investigate the graph parameter $γ(G)$, obtained by replacing the $\ell_2$-norm with the $\ell_\infty$-norm in the corresponding optimization problem. We establish a simple and explicit combinatorial formula expressing $γ(G)$ as the ratio of the order of the graph to its maximum transmission, thereby providing a direct graph-theoretic interpretation of the parameter. As a consequence, we obtain a polynomial-time algorithm based on breadth-first search, significantly simplifying the previously known linear programming approach. We prove that $γ(G)$ characterizes graph connectivity and completely characterize all $\ell_\infty$-Fiedler vectors as the vectors \[ \left\{\pm\left(1-γ(G)d(u,\cdot)\right):u\in \mathcal{M}(G)\right\}, \] where $\mathcal{M}(G)$ denotes the set of vertices of maximum transmission. Furthermore, we derive bounds for $γ(G)$ in terms of several classical graph invariants, including the distance spectral radius, Wiener index, algebraic connectivity, and Cheeger constant. Finally, we establish a product formula for $γ(G)$ under Cartesian products of graphs, leading to explicit expressions for important graph families such as hypercubes, Hamming graphs, grid graphs, and torus graphs.

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arXiv2026

Localized Turán-type inequalities for $Q$-index

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 \[ q(G) \leq 2n\left(1-\frac{1}{ω(G)}\right), \] where $ω(G)$ denotes the clique number of $G$. We first give a short algebraic proof of this result. For a vertex $v\in V(G)$, let $c(v)$ denote the order of the largest clique of $G$ containing $v$. Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: \[ q(G) \leq 2\sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right). \] Equality holds precisely for complete bipartite graphs when $ω(G)=2$, and for regular complete $ω(G)$-partite graphs when $ω(G)\geq 3$. As a consequence, we also obtain an analogous localized inequality for the $A_α$-matrix of $G$. Finally, we generalize the above localized inequality to vertex-weighted signed graphs. This contributes to the localization program for spectral Turán-type results.

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arXiv2026

Structural and extremal properties of $l_1$-Fiedler value

The algebraic connectivity $a(G)$, defined as the second smallest eigenvalue of the Laplacian matrix $L(G)$, admits a well-known variational characterization involving the minimization of a quadratic form subject to an $\ell_{2}$-norm constraint. In a recent work, Andrade and Dahl (2024) proposed an analogous formulation based on the $\ell_{1}$-norm, leading to the introduction of a new graph parameter $b(G)$, referred to as the $l_1$-Fiedler value. In this article, we undertake a detailed investigation of the structural and extremal properties of $b(G)$. We first derive a Nordhaus--Gaddum type inequality for $b(G)$. For trees, we determine both global maximizer and minimizers of $b(G)$, and present extremal constructions for trees with prescribed diameter, maximum degree, and number of pendant vertices. We further establish a connection between $b(G)$ and Laplacian matrices, and obtain a bound for $b(G)$ in terms of the edge connectivity, along with a complete characterization of the graphs attaining equality. We derive an explicit formula that describes the behaviour of $b(G)$ under the addition of pendant vertices. We also investigate the connection between $b(G)$ and the isoperimetric number.

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arXiv2025

Localization of spectral Turán-type theorems

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. $e$). Denote the adjacency eigenvalues of $G$ by $λ_1 \ge \cdots \ge λ_n$. We study localized refinements of spectral Turán-type theorems by replacing global parameters such as the clique number $ω(G)$, size $m$ and order $n$ of $G$ with local quantities $c(v)$ and $c(e)$. Motivated by a conjecture of Elphick, Linz and Wocjan (2024), we first propose a vertex-localized strengthening of Wilf's inequality: \[ \sqrt{s^{+}(G)} \le \sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right), \] where $s^+(G) = \sum_{λ_i > 0}λ_i^2$. Inspired by the Bollobás-Nikiforov conjecture (2007) on the first two eigenvalues, we then introduce an edge-localized analogue: \[λ_1^2(G) + λ_2^2(G) \le \sum_{e\in E(G)} 2\left(1-\frac{1}{c(e)}\right).\] As evidence of their validity, we verify the above conjectures for diamond-free graphs and random graphs. We also propose strengthening of the spectral versions of the Erdős, Stone and Simonovits Theorem by replacing the spectral radius with $\sqrt{s^{+}(G)}$ and establish it for all $F$-free graphs with $χ(F)=3$. A key ingredient in our proofs is a general upper bound relating $\sqrt{s^{+}(G)}$ to the triangle count $t(G)$. Finally, we prove a localized version of Nikiforov's walk inequality and conjecture a stronger localized version. These results contribute to the broader program of localizing spectral extremal inequalities.

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arXiv2025

When Are Standard Graph Products Isomorphic?

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected graphs for which their corresponding products are isomorphic. As a by-product, we identify a novel family of non-distance-regular graphs that possess fewer than d+1 distinct distance eigenvalues, where d represents the diameter of the graph. This result offers a new perspective on Problem 4.3 posed in [2], moving beyond the current approaches.

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arXiv2024

Constructing cospectral graphs by unfolding non-bipartite graphs

In 2010, Butler introduced the unfolding operation on a bipartite graph to produce two bipartite graphs, which are cospectral for the adjacency and the normalized Laplacian matrices. In this article, we describe how the idea of unfolding a bipartite graph with respect to another bipartite graph can be extended to nonbipartite graphs. In particular, we describe how unfoldings involving reflexive bipartite, semi-reflexive bipartite, and multipartite graphs are used to obtain cospectral nonisomorphic graphs for the adjacency matrix.

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arXiv2023

Bounds and extremal graphs for the energy of complex unit gain graphs

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the gain function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(Φ) $ is defined canonically. The energy $ \mathcal{E}(Φ) $ of a $ \mathbb{T} $-gain graph $ Φ$ is the sum of the absolute values of all eigenvalues of $ A(Φ) $. For any connected triangle-free $ \mathbb{T} $-gain graph $ Φ$ with the minimum vertex degree $ δ$, we establish a lower bound $ \mathcal{E}(Φ)\geq 2δ$ and characterize the equality. Then, we present a relationship between the characteristic and the matching polynomial of $ Φ$. Using this, we obtain an upper bound for the energy $ \mathcal{E}(Φ)\leq 2μ\sqrt{2Δ_e+1} $ and characterize the classes of graphs for which the bound sharp, where $ μ$ and $ Δ_e$ are the matching number and the maximum edge degree of $ Φ$, respectively. Further, for any unicyclic graph $ G $, we study the gains for which the gain energy $ \mathcal{E}(Φ) $ attains the maximum/minimum among all $ \mathbb{T} $-gain graphs defined on $G$.

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arXiv2023

A note on the distance and distance signless Laplacian spectral radius of complements of trees

In this article, we show that the generalized tree shift operation increases the distance spectral radius, distance signless Laplacian spectral radius, and the $D_α$-spectral radius of complements of trees. As a consequence of this result, we correct an ambiguity in the proofs of some of the known results.

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arXiv2023

On the spectrum of complex unit gain graph

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called $\mathbb{T}$-gain adjacency matrix. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. In this article, we study the cospectrality of matrices in $\mathbb{T}_{G} $ and we establish equivalent conditions for a graph $G$ to be a tree in terms of the spectrum and the spectral radius of matrices in $\mathbb{T}_{G} $. We identify a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G \in \mathfrak{F^{'}}$, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of $\mathbb{T}$-gain adjacency matrices on $ G \in \mathfrak{F^{'}} $ in terms of their largest eigenvalues. Thereupon, we characterize $\mathbb{T}$-gain graphs for which the spectral radius of the associated $\mathbb{T}$-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs. J. Graph Theory 85 (2017), no. 1, 217-248.].

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arXiv2023

Signed spectral Turań type theorems

A signed graph $Σ= (G, σ)$ is a graph where the function $σ$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $Σ$, denoted by $A(Σ)$, is defined canonically. In a recent paper, Wang et al. extended the eigenvalue bounds of Hoffman and Cvetković for the signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of the unsigned graphs. We extend the Turán's inequality for the signed graphs. Besides, we study the Bollobás and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph.

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arXiv2023

Extremal problems for the eccentricity matrices of complements of trees

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement.

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arXiv2022

Minimizers for the energy of eccentricity matrices of trees

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arXiv2022

Squared distance matrices of trees with matrix weights

Let $T$ be a tree on $n$ vertices whose edge weights are positive definite matrices of order $s$. The squared distance matrix of $T$, denoted by $Δ$, is the $ns \times ns$ block matrix with $Δ_{ij}=d(i,j)^2$, where $d(i,j)$ is the sum of the weights of the edges in the unique $(i,j)$-path. In this article, we obtain a formula for the determinant of $Δ$ and find $Δ^{-1}$ under some conditions.

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arXiv2022

On the construction of cospectral nonisomorphic bipartite graphs

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using partitioned tensor product. This extends the construction of Ji, Gong, and Wang \cite{ji-gong-wang}. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction \cite{godsil-mckay-1976}, and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al.

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arXiv2022

On the eccentricity matrices of trees: Inertia and spectral symmetry

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues} of $G$. In this article, we find the inertia of the eccentricity matrices of trees. Interestingly, any tree on more than $4$ vertices with odd diameter has two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the structure of the tree). A tree with even diameter has the same number of positive and negative $\mathcal{E}$-eigenvalues, which is equal to the number of 'diametrically distinguished' vertices (see Definition 3.1). Besides we prove that the spectrum of the eccentricity matrix of a tree is symmetric with respect to the origin if and only if the tree has odd diameter. As an application, we characterize the trees with three distinct $\mathcal{E}$-eigenvalues.

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