Comparison Between Two Eccentricity-based Topological Indices of Graphs

For a connected graph (G), the eccentric connectivity index (ECI) and the first Zagreb eccentricity index of (G) are defined as ( xi ^{c}(G)= sum_{v_i in V(G)}deg_G(v_i)varepsilon_G(v_i)) and (E_1(G)=sum_{v_iin V(G)}varepsilon_{G}(v_i)^{2}), respectively, where (deg_G(v_i)) is the degree of (v_i) in (G) and (varepsilon_G(v_i)) denotes the eccentricity of vertex (v_i )in (G). In this paper we compare the eccentric connectivity index and the first Zagreb eccentricity index of graphs. It is proved that (E_1(T)>xi^c(T)) for any tree (T). This improves a result by Das[25] for the chemical trees. Moreover, we also show that there are infinite number of chemical graphs (G) with (E_1(G)>xi^c(G)). We also present an example in which infinite graphs (G) are constructed with (E_1(G)=xi^c(G)) and give some results on the graphs (G) with (E_1(G)<xi^c(G)). Finally, an effective construction is proposed for generating infinite graphs with each comparative inequality possibility between these two topological indices

The general position number of the Cartesian product of two trees

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees

New transmission irregular chemical graphs

The transmission of a vertex $v$ of a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is a transmission irregular graph. It is shown that for any odd number $n\geq 7$ there exists a transmission irregular chemical tree of order $n$. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge

The smallest Hosoya index of unicyclic graphs with given diameter

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let ${cal{U}}_{n,d}$ be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from ${cal{U}}_{n,d}$ minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hosoya index of unicyclic graphs is determined