‎the harmonic index of a connected graph $g$‎, ‎denoted by $h(g)$‎, ‎is‎ ‎defined as $h(g)=sum_{uvin e(g)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in g‎. ‎in this paper‎, ‎expressions for the harary indices of the‎ ‎join‎, ‎corona product‎, ‎cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

the wiener index is a graph invariant that has found extensive application in chemistry. inaddition to that a generating function, which was called the wiener polynomial, who’sderivate is a q-analog of the wiener index was defined. in an article, sagan, yeh and zhang in[the wiener polynomial of a graph, int. j. quantun chem., 60 (1996), 959969] attainedwhat graph operations do to the wiener po...

By a symmetric graph we mean a graph X which automorphism group acts transitively on the arcs of X. A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. Tutte [31, 32] showed that every finite symmetric cubic graph is s-regular for some s ≤ 5. It is well-known that there are precisely five symmetric cubic graphs of girth less than 6. All these graphs can be re...

We find closed form formulas for the Kemeny’s constant and Kirchhoff index cluster G1{G2} of two highly symmetric graphs G1, G2, in terms parameters original graphs. also discuss some necessary conditions a graph to be symmetric.

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...