The first two cacti with larger multiplicative eccentricity resistance-distance

Abstract

For a connected graph $G$, the multiplicative eccentricity resistance-distance $\xi^*_R(G)$ is defined as $\xi^*_R(G)=\sum_{\{x,y\}\subseteq V(G)}\varepsilon(x)\cdot\varepsilon(y)R_G(x,y),$ where $\varepsilon(\cdot)$ is the eccentricity of the corresponding vertex and $R_G(x,y)$ is the effective resistance between vertices $x$ and $y$. A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Let $Cat(n;t)$ be the set of cacti possessing $n$ vertices and $t$ cycles, where $0\leq t \leq \frac{n-1}{2}$. In this paper, we first introduce some edge-grafting transformations which will increase $\xi^*_R(G)$. As their applications, the extremal graphs with maximum and second-maximum $\xi^*_R(G)$-value in $Cat(n; t)$ are characterized, respectively.