arXiv subjects

Pratima Panigrahi

Publications and source records attributed to Pratima Panigrahi.

9 recordsLinked to original sources

arXiv2026

Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs

Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs $\mathcal{F}$, a graph is called $\mathcal{F}$-free if it contains no member of $\mathcal{F}$ as a subgraph. The theta graph $θ(l_1,\ldots,l_k)$ consists of $k$ internally disjoint paths of lengths $l_1,\ldots,l_k$ with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all $\{C_3,C_4\}$-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all $\{θ(1,2,2),θ(1,2,3)\}$-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.

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arXiv2026

$A_α$-Spectra of $Q$- and $T$-Join Graphs with Applications to Cospectral Constructions

For $α\in [0,1]$, the $A_α$-matrix of a graph $G$ is defined by $A_α(G) = αD(G) + (1- α) A(G)$, where $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal degree matrix of $G$, respectively. In this paper, we study the $A_α$-characteristic polynomials and $A_α$-spectra of graphs obtained via four recently introduced join operations, namely the $Q$-vertex join, $Q$-edge join, $T$-vertex join, and $T$-edge join, applied to two graphs $G_1$ and $G_2$. We derive explicit expressions for the $A_α$-characteristic polynomials of these constructions when the first factor graph is regular. Furthermore, we determine the complete $A_α$-spectra of these graphs in terms of the $A_α$-spectra of the factor graphs, particularly when the second factor graph is regular or complete bipartite. The significance of these results lies in the fact that they enable efficient computation of the $A_α$-spectra of large complex graphs arising from these joins, directly from the $A_α$-spectra of the smaller constituent graphs, without explicitly constructing and handling the complex $A_α$-matrices of those large graphs. Finally, as an application, we demonstrate how to construct infinitely many families of non-isomorphic graphs that are $A_α$-cospectral.

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arXiv2026

Characterizing tricyclic graphs with pendant vertices having largest $A_α$-spectral radius

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_α$-matrix of $G$ is defined as \begin{equation*} A_α(G) = αD(G) + (1- α) A(G), \text{ for any } α\in [0,1]. \end{equation*} The $A_α$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_α(G)$. A tricyclic graph of order $n$ is a simple connected graph with $n+2$ edges. In this paper, we characterize the unique graph having the largest $A_α$-spectral radius for $α\in [\frac{1}{2}, 1)$ among all tricyclic graphs of order $n$ with $k (\geq 1)$ pendant vertices. As an application, we derive a sufficient spectral condition (alternate to the edge condition) to guarantee the absence of the tricyclic structure in a graph with $k$ pendant vertices.

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arXiv2026

Exploring new upper and lower bounds for the $A_α$-energy of graphs

Let $G$ be a graph on $n$ vertices and $m$ edges. For $α\in [0,1]$, the $A_α$-matrix of $G$ is defined as $A_α(G) = αD(G) + (1- α) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $ρ_1 \geq ρ_2 \ldots \geq ρ_n$ are the eigenvalues of $A_α(G)$, the $A_α$-energy of $G$ is defined as $E_{A_α}(G) = \sum_{i=1}^{n} |ρ_i -\frac{2αm}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_α}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_α}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph.

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arXiv2025

A method to optimize antipodal coloring span of graphs and its application

In this article, we study radio $k$-colorings of simple connected graphs $G$ with diameter $d$, where a radio $k$-coloring $g$ assigns non-negative integers to $V(G)$ (vertices of $G$) such that $|g(u) - g(v)| \geq 1 + k - d(u, v)$ for any two vertices $u, v$ with $1 \leq k \leq d$. The span of a radio $k$-coloring $g$, expressed by $rc_k(g)$, is the maximum integer assigned by $g$, and the radio $k$-chromatic number $rc_k(G)$ is the minimum span among all radio $k$-colorings of $G$. A coloring $g$ is minimal if $rc_k(g) = rc_k(G)$. When $k = d-1$, this coloring is known as the antipodal coloring, and $rc_{d-1}(G)$ referred to as the antipodal number, is denoted by $ac(G)$. We derive a sufficient condition for an antipodal coloring to be minimal and apply this criterion to determine the antipodal number of the generalized Petersen graph $GP(n,1)$ for all $n$ except when $n \equiv 2 \pmod{8}$, and for toroidal grids $T_{r,s} = C_r \square C_s$ when $rs$ is even. Additionally, we establish a lower bound for $ac(T_{r,s})$ when $rs$ is odd.

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arXiv2024

A new generalization of Fielder's lemma with applications

Very recently Ma and Wu \cite{wu2024generalization} obtained a generalization of Fielder's lemma and applied to find adjacency, Laplacian, and signless Laplacian spectra of $P_n-$ product of commuting graphs. In this paper, we give a generalization of Fielder's lemma applying which not only one gets generalized result in \cite{wu2024generalization} as a particular case, but also one can find several kind of spectra of $H$-product of graphs when $H$ is an arbitrary graph. Moreover, we compute adjacency spectrum of $H-$ product of commuting graphs and universal adjacency spectrum of $H-$ product of commuting regular graphs.

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arXiv2024

Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups

The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we determine the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases.

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arXiv2023

Universal adjacency spectrum of (proper) power graphs and their complements on some groups

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple graph $H$ with adjacency matrix $A(H)$ and degree diagonal matrix $D(H)$, the universal adjacency matrix is $U(H)= αA(H)+βD(H)+ γI +ηJ$, where $α(\neq 0), β, γ, η\in \mathbb{R}$, $I$ is the identity matrix and $J$ is the all-ones matrix of suitable order. One can study many graph-associated matrices, such as adjacency, Laplacian, signless Laplacian, Seidel etc. in a unified manner through the universal adjacency matrix of a graph. Here we study universal adjacency eigenvalues and eigenvectors of power graphs, proper power graphs and their complements on the group $\mathbb{Z}_n$, dihedral group ${D}_n$, and the generalized quaternion group ${Q}_n$. Spectral results of no kind for the complement of power graph on any group were obtained before. We determine the full spectrum in some particular cases. Moreover, several existing results can be obtained as very specific cases of some results of the paper.

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arXiv2021

On the distance spectrum of minimal cages and associated distance biregular graphs

A $(k,g)$-cage is a $k$-regular simple graph of girth $g$ with minimum possible number of vertices. In this paper, $(k,g)$-cages which are Moore graphs are referred as minimal $(k,g)$-cages. A simple connected graph is called distance regular(DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular(DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal $(k,g)$-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal $(k,g)$-cages we give a formula for distance spectral radius in terms of $k$ and $g$, and also determine polynomials of degree $[\frac{g}{2}]$, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal $(k,g)$-cage of diameter $d$ has $d+1$ distinct distance eigenvalues, and this partially answers a problem posed in [5]. We prove that every DBR graph is a $2$-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal $(k,g)$-cages. Finally we determine the full distance spectrum of subdivision of some minimal $(k,g)$-cages.

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