On Vertex-Edge and Edge-Vertex Connectivity Indices of Graphs
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Authors: S. PAWAR, A. M. NAJI, N. D. SONER, A. R. ASHRAFI AND A. GHALAVAND
DOI: 10.46793/KgJMat2402.225P
Abstract:
Let G be a graph with vertex set V (G) and edge set E(G). The vertex-edge degree of the vertex v, dGe(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, dGv(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, ϕv, and the edge-vertex connectivity index, ϕe, of a graph G were introduced. These are defined as ϕv(G) = ∑ v∈V (G)dGe(v)dG(v) and ϕe(G) = ∑ e=uv∈E(G)dG(e)dGv(e), where dG(v) is the degree of a vertex v ∈ V (G) and dG(e) is the number of edges in E(G) that are adjacent to e. In this paper, we will study the main properties of ϕv(G), ϕe(G) and establish some upper and lower bounds for them. The numbers ϕv and ϕe for titania nanotubes are also computed.
Keywords:
Vertex-edge degree, edge-vertex degree, vertex-edge connectivity index, edge-vertex connectivity index.
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