On Vertex-Edge and Edge-Vertex Connectivity Indices of Graphs

Shiladhar Pawar ; Ahmed Mohsen Naji ; Nandappa D. Soner ; Ali Reza Ashrafi ; Ali Ghalav

Kragujevac Journal of Mathematics, Tome 48 (2024) no. 2, p. 225

Résumé

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The vertex-edge degree of the vertex $v$, $d^e_{G}(v),$ equals to the number of different edges that are incident to any vertex from the open neighborhood of $v$. Also, the edge-vertex degree of the edge $e=uv$, $d_G^v(e)$, equals to the number of vertices of the union of the open neighborhood of $u$ and $v$. In this paper, the vertex-edge connectivity index, $\phi_v$, and the edge-vertex connectivity index, $\phi_e$, of a graph $G$ were introduced. These are defined as $\phi_v(G)=\sum_{v\in V(G)}d_G^e(v)d_G(v)$ and $\phi_e(G)=\sum_{e=uv\in E(G)}d_G(e)d_G^v(e)$, where $d_G(v)$ is the degree of a vertex $v \in V(G)$ and $d_G(e)$ is the number of edges in $E(G)$ that are adjacent to $e$. In this paper, we will study the main properties of $\phi_v(G)$, $\phi_e(G)$ and establish some upper and lower bounds for them. The numbers $\phi_v$ and $\phi_e$ for titania nanotubes are also computed.

Détail

Classification : 05C09, 05C07, 05C35
Keywords: vertex-edge degree, edge-vertex degree, vertex-edge connectivity index, edge-vertex connectivity index