arXiv subjects

Tahir Shamsher

Publications and source records attributed to Tahir Shamsher.

5 recordsLinked to original sources

arXiv2026

A maximum matching based refinement of Brouwers conjecture

Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $μ_1\geq \cdots \geq μ_{n-1}\geq μ_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}μ_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the sum of the largest Laplacian eigenvalues of graphs, {\em Discrete Appl. Math.}, 170:95--103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \leq k \leq \lfloor g/5 \rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, {\em Linear Algebra Appl.}, 557:327--338, (2018)], who established that the conjecture holds for $1 \leq k \leq \lfloor g/4 \rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\geq \lfloor g/2\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$ holds whenever $\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$, where $\overline{G}$ denotes the complement of $G$.

math.CO↗

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.

math.CO↗

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.

math.CO↗

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.

math.CO↗

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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