arXiv subjects

Akash Kalita

Publications and source records attributed to Akash Kalita.

4 recordsLinked to original sources

arXiv2026

Laplacian Pair State Transfer on Total Graphs

The total graph of a graph $G$, denoted $\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$, such that two vertices of $\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are either adjacent or incident. In this paper, we investigate the existence of Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $G$ is Laplacian integral, $r \geq 3$, and $r+1$ is not a Laplacian eigenvalue of $G$, then $\mathcal{T}(G)$ does not exhibit Laplacian perfect pair state transfer. In addition, we prove that under some mild conditions, $\mathcal{T}(G)$ exhibits Laplacian pretty good pair state transfer, where $r \geq 3$ and $r+1$ is not a Laplacian eigenvalue of $G$. Using these conditions, we obtain several infinite families of total graphs exhibiting Laplacian pretty good pair state transfer that fail to exhibit Laplacian perfect pair state transfer. We also prove that the total graph of the complete graph $K_n$ exhibits Pair-LPGST if and only if $n=3$.

math.CO↗

arXiv2025

State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs

The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1 \leq u \leq n-1$ and $\gcd(u,n) = 1$. In this paper, we classify all $X_n$ admitting pretty good fractional. We also classify all $X_n$ that admit fractional revival. It turns out that $X_n$ admits fractional revival if and only if it admits pretty good fractional revival. Further, we classify all $G_n$ admitting periodicity. As a consequence, we obtain all $G_n$ admitting perfect state transfer. We also classify $G_n$ admitting pretty good state transfer, pretty good fractional revival and fractional revival.

math.CO↗

arXiv2025

Pretty good fractional revival on abelian Cayley graphs

Let $\Gamma$ be a graph with the adjacency matrix $A$. The transition matrix of $\Gamma$, denoted $H(t)$, is defined as $H(t) := \exp(-\textbf{i}tA)$, where $\textbf{i} := \sqrt{-1}$ and $t$ is a real variable. The graph $\Gamma$ is said to exhibit fractional revival (FR in short) between the vertices $a$ and $b$ if there exists a positive real number $t$ such that $H(t){\textbf{e}_{a}} = \alpha{\textbf{e}_{a}} + \beta{\textbf{e}_{b}}$, where $\alpha, \beta \in \mathbb{C}$ such that $\beta \neq 0$ and $|\alpha|^2 + |\beta|^2 = 1$. The graph $\Gamma$ is said to exhibit pretty good fractional revival (PGFR in short) between the vertices $a$ and $b$ if there exists a sequence of real numbers $\{t_k\}$ with $\lim_{k\to\infty} H(t_k){\textbf{e}_{a}} = \alpha{\textbf{e}_{a}} + \beta{\textbf{e}_{b}}$, where $\alpha, \beta \in \mathbb{C}$ such that $\beta \neq 0$ and $|\alpha|^2 + |\beta|^2 = 1$. In the definition of PGFR, if $\alpha=0$ then $\Gamma$ is said to exhibit pretty good state transfer (PGST in short) between $a$ and $b$. In this paper, we obtain some sufficient conditions for circulant graphs exhibiting PGFR. We also find some sufficient conditions for non-circulant abelian Cayley graphs exhibiting PGFR. From these sufficient conditions, we find infinite families of circulant graphs and non-circulant abelian Cayley graphs exhibiting PGFR that fail to exhibit FR and PGST. Finally, we obtain some necessary conditions for some families of circulant graphs exhibiting PGFR. Some of our results generalize the results of Chan et al. [Pretty good quantum fractional revival in paths and cycles. \textit {Algebr. Comb.} 4(6) (2021), 989-1004.] for cycles.

math.CO↗

arXiv2024

Perfect state transfer on Cayley graphs over a non-abelian group of order $8n$

The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$, where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n}, S)$. We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n}, S)$.

quant-ph↗