Molecular descriptors are mathematical representations of molecular properties, generated through numerous algorithms. These numerical values are used to quantitatively represent the physical and ...chemical attributes of molecules. In the field of chemical graph theory, two indices, namely the revised Szeged index and the revised edge-Szeged index, were introduced to characterize molecular properties. The Szeged index Sz(Γ) of a simple connected graph Γ is computed by summing the products of n e u( ) and n e v ( ) for all edges e uv = in Γ, where n e u( ) denotes the number of vertices in Γ that are closer to vertex u than to vertex v, and n e v ( ) is defined similarly. In this paper, the role of different variants of Szeged indices in modeling different physical properties of alkanes and benzenoid hydrocarbon is investigated. Their isomer discrimination ability is also examined. In addition, we obtain lower and upper bounds on revised Szeged index, revised edge-Szeged index and the difference between vertex-edge Szeged index and edge-vertex Szeged index of bicyclic graphs.
Let W(G),Sz(G) and Sz∗(G) be the Wiener index, Szeged index and revised Szeged index of a connected graph G, respectively. Call Ln,r a lollipop if it is obtained by identifying a vertex of Cr with an ...end-vertex of Pn−r+1. For a connected unicyclic graph G with n≥4 vertices, Hansen et al. (2010) conjectured: (A)Sz(G)W(G)≤2−8n2+7,if n is odd,2,if n is even,(B)Sz∗(G)W(G)≥1+3(n2+4n−6)2(n3−7n+12),if n≤9,1+24(n−2)n3−13n+36,if n≥10,(C)Sz∗(G)W(G)≤2+2n2−1,if n is odd,2,if n is even,where the equality in (A) holds if and only if G is the lollipop Ln,n−1 if n is odd, and the cycle Cn if n is even; the equality in (B) holds if and only if G is the lollipop Ln,3 if n≤9, and Ln,4 if n≥10, whereas the equality in (C) holds if and only if G is the cycle Cn. In this paper, we not only confirm these conjectures but also determine the lower bound of Sz∗(G)∕W(G) (resp. Sz(G)∕W(G)) for cyclic graphs G. The extremal graphs that achieve these lower bounds are characterized.
The Szeged index Sz(G) of a simple connected graph G is the sum of the terms n
u
(e)n
v
(e) over all edges e = uv of G, where n
u
(e) is the number of vertices of G lying closer to u than v, and n
v
...(e) is defined analogously. The aim of this paper is to present some relationship between Szeged index and some of its variants such as the edge-vertex Szeged index, the vertex-edge Szeged index and revised Szeged index. Moreover, we obtain lower and upper bounds on the difference between vertex-edge Szeged index and edge-vertex Szeged index of unicyclic graphs.
For a given graph G, its edge Szeged index is denoted by Sze(G)=∑e=uv∈E(G)mu(e)mv(e), where mu(e) and mv(e) are the number of edges in G with distance to u less than to v and the number of edges in G ...further (distance) to u than v, respectively. In the paper, the bounds of edge Szeged index on bicyclic graphs are determined. Furthermore, the graphs that achieve the bounds are completely characterized.
The edge-Szeged index of a graph G is defined as Sze(G)=∑uv∈E(G)mu(uv|G)mv(uv|G), where mu(uv|G) (resp., mv(uv|G)) is the number of edges whose distance to vertex u (resp., v) is smaller than the ...distance to vertex v (resp., u), respectively. In this paper, we characterize the graphs with minimum edge-Szeged index among all the unicyclic graphs with given order and perfect matchings.
•The variable Szeged index exceeds the variable Wiener index for exponents larger than 1.•The critical exponent achieving equality is not always unique.•The critical exponent is unique for bipartite ...graphs, block graphs and sparse graphs.•The critical exponent is unique with high probability for random graphs of any density.
We resolve two conjectures of Hriňáková et al. (2019)10 concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in (0,1, below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding 1. They proved the weak conjecture for bipartite graphs, and the strong conjecture for trees.
In this note we disprove the strong conjecture, although we show that it is true for almost all graphs, and for bipartite and block graphs. We also show that the weak conjecture holds for all graphs by proving a majorization relationship.
Szeged index of hollow hexagons Lopera, Carolina; Cruz, Roberto
International journal of quantum chemistry,
12/2023, Volume:
123, Issue:
23
Journal Article
Peer reviewed
Open access
Abstract
The Szeged index of a connected graph () is a well known distance based topological index. A primitive coronoid system is a coronoid system formed by a single chain in a macro‐cyclic ...arrangement consisting of linearly and angularly annelated hexagons. The angular hexagons are called corners. A hollow hexagon is a primitive coronoid system with exactly six corners. In this paper we calculate the values of Szeged index of hollow hexagons using the cut method.
Szeged and Mostar root-indices of graphs Brezovnik, Simon; Dehmer, Matthias; Tratnik, Niko ...
Applied mathematics and computation,
04/2023, Volume:
442
Journal Article
Peer reviewed
Open access
•We introduce several Szeged and Mostar root-indices of graphs.•The root-indices are defined as unique positive roots of modified graph polynomials.•Various analytical results of root-indices are ...derived.•Discrimination, correlations, structure sensitivity, and abruptness are calculated.•The obtained numerical results are compared with the already known similar descriptors.
Various distance-based root-indices of graphs are introduced and studied in the present article. They are obtained as unique positive roots of modified graph polynomials. In particular, we consider the Szeged polynomial, the weighted-product Szeged polynomial, the weighted-plus Szeged polynomial, and the Mostar polynomial. We derive closed formulas of these polynomials for some basic families of graphs. Consequently, we provide closed formulas for some root-indices and examine the convergence of sequences of certain root-indices. Moreover, some general properties of studied root-indices are stated. Finally, numerical results related to discrimination power, correlations, structure sensitivity, and abruptness of root-indices are calculated, interpreted, and compared to already known similar descriptors.
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