Let Sz(G) and W (G) be the revised Szeged index and the Wiener index of a graph G. Chen, Li, and Liu [European J. Combin. 36 (2014) 237–246] proved that if G is a non-bipartite connected graph of order n ≥ 4, then Sz(G) −W (G) ≥ ( n + 4n− 6 ) /4. Using a matrix method we prove that if G is a non-bipartite graph of order n, size m, and girth g, then Sz(G)−W (G) ≥ n ( m− 3n 4 ) + P (g), where P i...

During the search for new structural descriptors we have defined the information-theory operators U(M), V(M), X(M), and Y(M), that are computed from atomic invariants and measure the information content of the elements of molecular matrices. Structural descriptors computed with these four information-theory operators are used to develop structure-property models for the boiling temperature, mol...

A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. A new counting polynomial, called the "Omega" W(G, x) polynomial, was recently proposed by Diudea on the ground of quasi-orthogonal cut "qoc" edge strips in a polycyclic graph. In this paper, the vertex PI, Szeged and omega polynomials of carbon nanocones CNC4[n] are computed.

Let G be a connected graph and ξ(G) = Sze(G)−We(G), where We(G) denotes the edge Wiener index and Sze(G) denotes the edge Szeged index of G. In an earlier paper, it is proved that if T is a tree then Sze(T ) = We(T ). In this paper, we continue our work to prove that for every connected graph G, Sze(G) ≥ We(G) with equality if and only if G is a tree. We also classify all graphs with ξ(G) ≤ 5. ...

Let W (G) and Sz(G) be the Wiener index and the Szeged index of a connected graph G. It is proved that if G is a connected bipartite graph of order n ≥ 4, size m ≥ n, and if ` is the length of a longest isometric cycle of G, then Sz(G) − W (G) ≥ n(m − n + ` − 2) + (`/2) − ` + 2`. It is also proved if G is a connected graph of order n ≥ 5 and girth g ≥ 5, then Sz(G) − W (G) ≥ PIv(G) − n(n − 1) +...

It is shown that the graphs for which the Szeged index equals ‖G‖·|G| 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables to disprove a conjecture proposed in [Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of ...

Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On other hand, polynomials related to these were also introduced, for example Szeged polynomial, edgeSzeged PI etc. In this paper, we introduce a concept general polynomial connected strength-weighted graph. It turns out that includes all above mentioned infinitely ma...

Let $G$ be a simple connected graph. In this paper, Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Sze...