The Kronecker product of two connected graphs G1,G2, denoted by G1 × G2, is the graph with vertex set V (G1 ×G2) = V (G1)×V (G2) and edge set E(G1 ×G2) = {(u1, v1)(u2, v2) : u1u2 ∈ E(G1), v1v2 ∈ E(G2)}. The kth power Gk of G is the graph with vertex set V (G) such that two distinct vertices are adjacent in Gk if and only if their distance apart in G is at most k. A connected graph G is called s...

A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity λ′ of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each connected component has at least two vertices. A graph G is called λ′-optimal if λ′(G) = min{dG(u)+dG(v)−2 : uv is an edge in G}. This paper proves t...

We prove that the graph complex is a strong homotopy Lie (super) bialgebra.

Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zag...

The Randić index of a graph G, denoted by R(G), is defined as the sum 1/d(u)d(v) for all edges uv where d(u) denotes degree vertex u in G. In this note, we show that R(L(T))>n4 any tree T order n≥3. A number relevant conjectures are proposed.

The applications of super-resolution (SR) technology in the field image completion are successful. Nevertheless, industry demand not only but also topology and time-series completion. In this article, SR on a graph is studied scenario recovering measurements power distribution systems for cost saving security & stability improvement. flow voltage magnitude feeders reported at different frequenc...