The linear chain as an extremal value of VDB topological indices of polyomino chains

Abstract

We give conditions on the numbers {ϕ ij } under which a vertexdegree-based topological index T I of the form T I (G) = 1≤i≤j≤n−1 m ij ϕ ij , where G is a graph with n vertices and m ij is the number of ij-edges, has the linear chain as an extreme value among all polyomino chains. As a consequence, we deduce that over the polyomino chains, the linear chain has the maximal value of the Randić index, the sum-connectivity index, the harmonic index and the geometric-arithmetic index and the minimal value of the first Zagreb index and the second Zagreb index.

Extremal polyomino chains of VDB topological indices

“…In this paper we study the extremal values of VDB topological indices using transformations of polyomino chains, a technique which was successful in the study of extremal values of VDB topological indices over catacondensed hexagonal systems [11]. This is a natural continuation of the results obtained in ( [12], [13]), where conditions on the numbers {ϕ ij } were given in order to assure that the linear chain or the zig-zag chain are extremal values of the induced VDB topological index T . However, as we shall see in this paper, there are other polyomino chains that are extremal values of VDB topological indices.…”

“…non-positive) then the linear chain has maximal (resp. minimal) T -value over P n [12]. As a consequence of the values given in Table 2, it was deduced that the linear chain has extremal value for important topological indices.…”

“…If P is any polyomino chain, then we denote by |P | the number of squares P has. Recently several transformations of polyomino chains were introduced in order to determine when the linear chain and the zig-zag chain have extremal values of a VDB topological index over P n ([12], [13]). These transformations are classified as linearizing and angularizing transformations.…”

“…The linearizing transformations L 1 and L 2 are shown in Figure 3, where L is a linear subchain, X is a polyomino subchain and v a vertex with degree 3 or 4. The variation of a VDB topological index T induced by {ϕ ij } under the linearizing transformations is given as follows [12]:…”

Bond incident degree (BID) indices of polyomino chains: A unified approach

“…Equation √ a+b . Rada [25] recently proved that the linear chain L n has the extremal value for many well known topological indices (including the aforementioned indices). This result can also be deduced from Theorem 2.10: Proof.…”