arXiv subjects

S. Pirzada

Publications and source records attributed to S. Pirzada.

At least 19 recordsLinked to original sources

arXiv2026

Graphs with $\{P_3,P_4,P_5\}$-factors in terms of size and spectral radius

Let $G$ be a connected graph of order $n$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. In this paper, we establish a sufficient condition on the size of the graph $G$ with minimum degree $\delta$ to have a $\{P_3, P_4, P_5\}$-factor. Subsequently, we provide another sufficient condition on the adjacency spectral radius, ensuring that a connected graph $G$ with minimum degree $\delta$ contains a $\{P_3, P_4, P_5\}$-factor.

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arXiv2026

On distance integral and distance Laplacian integral graphs

Let $G$ be a connected graph on $n$ vertices and let $D(G)$ and $D^{L}(G)$ be the distance and the distance Laplacian matrices associated with $G$. A graph $G$ is said to be $D$-integral (resp. $D^L$-integral) if all eigenvalues of $D(G)$ (resp. $D^L(G)$) are integers. In this paper, we obtain various conditions under which the graphs $a\overline{K_m}\nabla C_n$ and $K_{p,p}\nabla C_n$ are distance integral. We also obtain conditions on $m$, $n$ under which the dumbbell graph $\boldsymbol{DB}(W_{m,n})$ is $D^L$-integral.

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arXiv2026

The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph

A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4, 3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is shown that $\lambda(G)\leq \frac{1+\sqrt{4m-3}}{2}$ holds for all $H(4,3)-$free graphs of odd size $m\geq 44,$ and the equality holds if and only if $G\cong S_{\frac{m+3}{2},2},$ where $S_{\frac{m+3}{2},2}$ is the $m-$edge book graph $K_2 \vee \frac{m-1}{2}K_1,$ where $K_2 \vee \frac{m-1}{2}K_1,$ denotes the join of $K_2$ and $\frac{m-1}{2}K_1.$ Let $\mathcal{G}(m,H(4,3))$ denote the family of $H(4,3)$-free graphs with $m$ edges and no isolated vertices. We write $ \mathcal{G}(m,H(4,3)) \setminus \left\{ K_2 \vee \tfrac{m-1}{2}K_1 \right\} $ for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over $\mathcal{G}(m,H(4,3))\setminus \{K_2 \vee \frac{m-1}{2}K_1\}$ for odd $m\geq 58$ and characterize the unique extremal graph attaining this bound.

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arXiv2025

On connected graphs with finite spectral redundancy index and Pythagorean triplets

This article investigates spectral redundancy, a concept initially introduced by Alberto Seeger. Spectral redundancy arises when different connected induced subgraphs of a graph share the same spectral radius in their adjacency spectrum. Let $b(G)$ denote the total number of non-isomorphic induced subgraphs of $G$, and $c(G)$ represents the cardinality of the set of spectral radius of all connected induced subgraphs of $G$. The spectral redundancy of a graph $ G $ is defined as the ratio $ \frac{b(G)}{c(G)} $. The supremum of this ratio across all graphs in a family is called the spectral redundancy index of that family. We focus on a family of graphs that exhibit spectral redundancy and we find out the spectral redundancy index of this family. Furthermore, we investigate the connection between the spectral redundancy of these graphs and the presence of Pythagorean triplets.

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arXiv2024

On the spectral redundancy of pineapple graphs

In this article, we explore the concept of spectral redundancy within the class of pineapple graphs, denoted as $\mathcal{P}(α,β)$. These graphs are constructed by attaching $β$ pendent edges to a single vertex of a complete graph $K_α$. A connected graph $G$ earns the title of being spectrally non-redundant if the spectral radii of its connected induced subgraphs remain distinct. Spectral redundancy, on the other hand, arises when there is a repetition of spectral radii among the connected induced subgraphs within $G$. Specifically, we analyze the adjacency spectrum of $\mathcal{P}(α,β)$, revealing distinct eigenvalues including $0$, $-1$, and additional eigenvalues, some negative and others positive. Our investigation focuses on determining the spectral redundancy within this class of graphs, shedding light on their unique structural properties and implications for graph theory.

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arXiv2024

On maximum spectral radius of $\{H(3,3),~H(4,3)\}$-free graphs

Let $G$ be a simple connected graph of size $m$. Let $A$ be the adjacency matrix of $G$ and let $ρ(G)$ be the spectral radius of $G$. A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. Let $H(\ell,3)$ be the graph formed by taking a cycle of length $\ell$ and a triangle on a common vertex. Recently, Li, Lu and Peng [Y. Li, L. Lu, Y. Peng, Spectral extremal graphs for the bowtie, Discrete Math. 346(12) (2023) 113680.] showed that the unique $m$-edge $H(3,3)$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$ and the condition $m\ge 8$ is tight. In particular, if $G$ does not contain $H(3,3)$ as induced subgraph, they proved that $ρ(G) \leq \frac{1+\sqrt{4m-3}}{2} $ and equality holds when $G$ is isomorphic to $S_{\frac{m+3}{2},2}$. Note that Li et al. denoted $H(3,3)$ by $F_2$. In this paper, we find the maximum spectral radius and identify the graph with the largest spectral radius among all \{$H(3,3), H(4,3)$\}-free graphs of size odd $m$, where $m\geq 259$. Coincidentally, we show that $ρ(G) \leq \frac{1+\sqrt{4m-3}}{2}$ when $G$ forbids both $H(3,3)$ and $H(4,3)$. In our case, the equality holds when $G$ is isomorphic to the same graph.

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arXiv2023

Direct sum graph of the subspaces of a finite dimensional vector space over finite fields

In this paper, we introduce a new graph structure, called the $direct~ sum ~graph$ on a finite dimensional vector space. We investigate the connectivity, diameter and the completeness of $Γ_{U\oplus W}(\mathbb{V})$. Further, we find its domination number and independence number. We also determine the degree of each vertex in case the base field is finite and show that the graph $Γ_{U\oplus W}(\mathbb{V})$ is not Eulerian. We also show that under some mild conditions the graph $Γ_{U\oplus W}(\mathbb{V})$ is triangulated. We determine the clique number of $Γ_{U\oplus W}(\mathbb{V})$ for some particular cases. Finally, we find the size, girth, edge-connectivity and the chromatic number of $Γ_{U\oplus W}(\mathbb{V})$.

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arXiv2023

On the Estrada index of unicyclic and bicyclic signed graphs

Let $Γ=(G, σ)$ be a signed graph of order $n$ with eigenvalues $μ_1,μ_2,\ldots,μ_n.$ We define the Estrada index of a signed graph $Γ$ as $EE(Γ)=\sum_{i=1}^ne^{μ_i}$. We characterize the signed unicyclic graphs with the maximum Estrada index. The signed graph $Γ$ is said to have the pairing property if $μ$ is an eigenvalue whenever $-μ$ is an eigenvalue of $Γ$ and both $μ$ and $-μ$ have the same multiplicities. If $Γ_{p}^-(n, m)$ denotes the set of all unbalanced graphs on $n$ vertices and $m$ edges with the pairing property, we determine the signed graphs having the maximum Estrada index in $Γ_{p}^-(n, m)$, when $m=n$ and $m=n+1$. Finally, we find the signed graphs among all unbalanced complete bipartite signed graphs having the maximum Estrada index.

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arXiv2023

Spectra of s-neighbourhood corona of two signed graphs

A signed graph $S=(G, σ)$ is a pair in which $G$ is an underlying graph and $σ$ is a function from the edge set to $\{\pm1\}$. For signed graphs $S_{1}$ and $S_{2}$ on $n_{1}$ and $n_{2}$ vertices, respectively, the signed neighbourhood corona $S_{1} \star_s S_{2}$ (in short s-neighbourhood corona) of $S_{1}$ and $S_{2}$ is the signed graph obtained by taking one copy of $S_{1}$ and $n_{1}$ copies of $S_{2}$ and joining every neighbour of the $i$th vertex of $S_{1}$ with the same sign as the sign of incident edge to every vertex in the $i$th copy of $S_{2}$. In this paper, we investigate the adjacency, Laplacian and net Laplacian spectrum of $S_{1} \star_s S_{2}$ in terms of the corresponding spectrum of $ S_{1}$ and $ S_{2}$. We determine $(i)$ the adjacency spectrum of $S_{1} \star_s S_{2}$ for arbitrary $S_{1} $ and net regular $ S_{2}$, $(ii)$ the Laplacian spectrum for regular $S_{1} $ and regular and net regular $ S_{2}$ and $(iii)$ the net Laplacian spectrum for net regular $S_{1} $ and arbitrary $ S_{2}$. As a consequence, we obtain the signed graphs with $4$ and $5$ distinct adjacency, Laplacian and net Laplacian eigenvalues. Finally, we show that the signed neighbourhood corona of two signed graphs is not determined by its adjacency (resp., Laplacian, net Laplacian) spectrum.

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arXiv2023

On the eigenvalues and energy of the $A_α$-matrix of graphs

For a graph $G$, the generalized adjacency matrix $A_α(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_α(G)=αD(G)+(1-α) A(G)$ for $0\leq α\leq 1$. This matrix has been found to be useful in merging the spectral theories of $A(G)$ and the signless Laplacian matrix $Q(G)$ of the graph $G$. The generalized adjacency energy or $A_α$-energy is the mean deviation of the $A_α$-eigenvalues of $G$ and is defined as $E(A_α(G))=\sum_{i=1}^{n}|p_i-\frac{2αm}{n}|$, where $p_i$'s are $A_α$-eigenvalues of $G$. In this paper, we investigate the $A_α$-eigenvalues of a strongly regular graph $G$. We observe that $A_α$-spectral radius $p_1$ satisfies $δ(G)\leq p_1 \leq Δ(G)$, where $δ(G)$ and $Δ(G)$ are, respectively, the smallest and the largest degrees of $G$. Further, we show that the complete graph is the only graph to have exactly two distinct $A_α$-eigenvalues. We obtain lower and upper bounds of $A_α$-energy in terms of order, size and extremal degrees of $G$. We also discuss the extremal cases of these bounds.

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arXiv2022

On distance Laplacian spread and Wiener index of a graph

Let $G$ be a simple connected simple graph of order $n$. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G), \partial_{2}^{L}(G),\dots,\partial_{n}^{L}(G)$. The \textit{ distance Laplacian spread} $DLS(G)$ of a connected graph $G$ is the difference between largest and second smallest distance Laplacian eigenvalues, that is, $\partial_{1}^{L}(G)-\partial_{n-1}^{L}(G)$. We obtain bounds for $DLS(G)$ in terms of the Wiener index $W(G)$, order $n$ and the maximum transmission degree $Tr_{max}(G)$ of $G$ and characterize the extremal graphs. We obtain two lower bounds for $DLS(G)$, the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected $ k-partite$ graph $G$, $k\leq n-1$, with $n$ vertices having disconnected complement, we show that $ DLS(G)\geq \Big \lfloor \frac{n}{k}\Big \rfloor$ with equality if and only if $G$ is a $complete ~ k-partite$ graph having cardinality of each independent class same and $n \equiv 0 \pmod k$.

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arXiv2022

On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $λ_1(RD^L(G))\geq λ_2(RD^L(G))\geq \dots \geq λ_n(RD^L(G))$. The largest eigenvalue $λ_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number of components in the complement graph $\bar{G}$ of $G$. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with $n$ vertices. Also, we show that the star graph $S_n$ is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with $n$ vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph $K_n$, $K_n-e$, the star $S_n$, the complete balanced bipartite graph $K_{\frac{n}{2},\frac{n}{2}}$ and the complete split graph $CS(n,α)$ are all determined from the $RD^L$-spectrum.

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arXiv2022

On adjacency and Laplacian cospectral non-isomorphic signed graphs

Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.

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arXiv2022

On Zagreb index, signless Laplacian eigenvalues and signless Laplacian energy of a graph

Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $A(G)$ and $D(G)$ denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of $G$. Let $q_1\geq q_2\geq \dots\geq q_n\geq 0$ be the signless Laplacian eigenvalues of $G$. The largest signless Laplacian eigenvalue $q_1$ is called the signless Laplacian spectral radius or $Q$-index of $G$ and is denoted by $q(G)$. Let $S^+_k(G)=\displaystyle\sum_{i=1}^{k}q_i$ and $L_k(G)=\displaystyle\sum_{i=0}^{k-1}q_{n-i}$, where $1\leq k\leq n$, respectively denote the sum of $k$ largest and smallest signless Laplacian eigenvalues of $G$. The signless Laplacian energy of $G$ is defined as $QE(G)=\displaystyle\sum_{i=1}^{n}|q_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average vertex degree of $G$. In this article, we obtain upper bounds for the first Zagreb index $M_1(G)$ and show that each bound is best possible. Using these bounds, we obtain several upper bounds for the graph invariant $S^+_k(G)$ and characterize the extremal cases. As a consequence, we find upper bounds for the $Q$-index and lower bounds for the graph invariant $L_k(G)$ in terms of various graph parameters and determine the extremal cases. As an application, we obtain upper bounds for the signless Laplacian energy of a graph and characterize the extremal cases.

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arXiv2022

On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$. The largest eigenvalue $\partial_{1}^{L}(G)$ is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity $n-3$. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity $n-4$ together with one of the distance Laplacian eigenvalue as $n$ of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue $n$ is of multiplicity $n-4$.

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arXiv2022

Distance Laplacian eigenvalues of graphs and chromatic and independence number

For a connected graph $G$ of order $n$, let $Diag(Tr)$ be the diagonal matrix of vertex transmissions and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $D^L(G)=Diag(Tr)-D(G)$ and the eigenvalues of $D^{L}(G)$ are called the distance Laplacian eigenvalues of $G$. Let $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$ be the distance Laplacian eigenvalues of $G$. Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L} } I$) be the number of distance Laplacian eigenvalues of $G$ which lie in the interval $I$. For a prescribed interval $I$, we determine $m_{D^{L} }I$ in terms of independence number $α(G)$, chromatic number $χ$, number of pendant vertices and diameter $d$ of the graph $G$. In particular, we prove that $m_{D^{L}(G) }[n,n+2)\leq χ-1$, ~$m_{D^{L}(G) }[n,n+α(G))\leq n-α(G)$ and we show that the inequalities are sharp. We also show that $m_{D^{L} (G )}\bigg( n,n+\left\lceil\frac{n}χ\right\rceil\bigg)\leq n- \left\lceil\frac{n}χ\right\rceil-C_{\overline{G}}+1 $, where $C_{\overline{G}}$ is the number of components in $\overline{G}$, and discuss some cases where the bound is best possible. In addition, we prove that $m_{D^{L} (G )}[n,n+p)\leq n-p$, where $p\geq 1$ is the number of pendant vertices. Also, we characterize graphs of diameter $d\leq 2$ which satisfy $m_{D^{L}(G) } (2n-1,2n )= α(G)-1=\frac{n}{2}-1$. At the end, we propose some problems of interest.

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arXiv2021

On distance Laplacian energy in terms of graph invariants

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ ρ^{L}_{1}\geq ρ^{L}_{2}\geq \cdots \geq ρ^{L}_{n}$, the distance Laplacian energy $DLE(G)$ is defined as $DLE(G)=\sum_{i=1}^{n}\left|ρ^{L}_i-\frac{2 W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $DLE(G)$ in terms of the order $n$, the Wiener index $W(G)$, independence number, vertex connectivity number and other given parameters. We characterize the extremal graphs attaining these bounds. We show that the complete bipartite graph has the minimum distance Laplacian energy among all connected bipartite graphs and complete split graph has the minimum distance Laplacian energy among all connected graphs with given independence number. Further, we obtain the distance Laplacian spectrum of the join of a graph with the union of two other graphs. We show that the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}), 1\leq t \leq \lfloor\frac{n-k}{2}\rfloor $, has the minimum distance Laplacian energy among all connected graphs with vertex connectivity $k$. We conclude this paper with a discussion on trace norm of a matrix and the importance of our results in the theory of trace norm of the matrix $D^L(G)-\frac{2W(G)}{n}I_n$.

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arXiv2021

On the eigenvalues of signed complete bipartite graphs

Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. The adjacency matrix of $Γ=(G, σ)$ is a square matrix $A(Γ)=A(G, σ)=\left(a_{i j}^σ\right)$, where $a_{i j}^σ=σ\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},σ)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p=\{u_1,u_2,\ldots,u_p\}$ and $V_q=\{v_1,v_2,\ldots,v_q\}$. Let $(K_{p, q},σ)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, which contains all negative edges of the signed graph $(K_{p, q},σ)$. We show that the multiplicity of eigenvalue $0$ in $(K_{p, q},σ)$ is at least $ p+q-2k-2$, where $k=min(r,s)$. We determine the spectrum of signed complete bipartite graph whose negative edges induce disjoint complete bipartite subgraphs and path. We obtain the spectrum of signed complete bipartite graph whose negative edges (positive edges) induce an $r-$ regular subgraph $H$. We find a relation between the eigenvalues of this signed complete bipartite graph and the non-negative eigenvalues of $H$.

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