Spectral analysis of eccentricity matrix for one-point-corona of graphs

Among various graph operations, the corona product stands out as one of the well-known and extensively analyzed due to its elegant structural properties. Over the time, numerous variants of the corona operation have been introduced and are widely studied. In this paper, we investigate the spectral properties of the eccentricity matrix for one such variant. Let $G$ and $H$ be two connected graphs on $m$ and $n$ vertices, respectively. Let $H$ be rooted at a designated vertex $z$. One-point-corona, $G{\circ }_{z}H$ is obtained by taking one copy of $H$ for each vertex of $G$ and joining the root vertex $z$ in each copy of $H$ to the corresponding vertex in $G$ by an edge. In this article, we study the eccentricity matrix, $\varepsilon$, of one-point-corona of two graphs. First, we characterize the irreducibility of $\varepsilon \left(G{\circ }_{z}H\right)$. We prove this result by demonstrating the connectedness of the direct product of two graphs when one of them has a self-loop. Under the assumption that $G$ is self-centered, we investigate the $\varepsilon$-spectrum for $G{\circ }_{z}H$ and identify the collection of $\varepsilon$-cospectral graphs and extremal graphs. In this process of finding the extremal graphs in terms of their $\varepsilon$-spectral radius, we establish a stronger result of organizing the graphs, $G{\circ }_{z}H$, in a linear ordering of their spectral radius when $G$ is self-centered and $H$ is any rooted graph whose root vertex has a fixed eccentricity.