Connected vertex–Edge dominating sets and connected vertex–Edge domination polynomials of triangular ladder

Abstract

Let $G$ be a simple connected graph of order $n$. Let $D_{\text {cye }}(G, i)$ be the family of connected vertex-edge dominating sets of $G$ with cardinality $i$. The polynomial
$$
D_{\text {cve }}(G, x)=\sum_{i=\gamma_{\text {cve }}(G)}^n d_{\text {cve }}(G, i) x^i
$$
is called the connected vertex - edge domination polynomial of $G$ where $d_{\text {cve }}(G, i)$ is the number of vertex edge dominating sets of $G$. In this paper, we study some properties of connected vertex - edge domination polynomials of the Triangular Ladder $T L_n$. We obtain a recursive formula for $d_{\text {cve }}\left(T L_{n, i}\right)$. Using this recursive formula, we construct the connected vertex - edge domination polynomial
$$
D_{\text {cve }}\left(T L_{n, x}\right)=\sum_{i=n-2}^{2 n} d_{c v e}\left(T L_{n, i}\right) x^i
$$
of $T L_n$, where $D_{\text {cve }}\left(T L_{n, i}\right)$ is the number of connected vertex - edge dominating sets of $T L_n$ with cardinality $i$ and some properties of this polynomial have been studied.