Let $G$ be a strongly connected digraph with vertex set $\{v_1, v_2, \dots, v_n\}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of matrix were proposed Yan Yeh (Adv. Appl. Math.), Bapat et al. (Linear Algebra Appl.) independently, one is $q$-distance matrix, other exponential matrix. Given nonzero indeterminate $q$, $\mathscr{D}_G=(\mathscr{D}_{...

Graham, Hoffman and Hosoya gave a very nice formula about the determinant of the distance matrix DG of a graph G in terms of the distance matrix of its blocks. We generalize this result to a q-analogue of DG. Our generalization yields results about the equality of the determinant of the mod-2 (and in general mod-k) distance matrix (i.e. each entry of the distance matrix is taken modulo 2 or k) ...

‎this paper obtains the exact solutions of the wave equation as a second-order partial differential equation (pde)‎. ‎we are going to calculate polynomial and non-polynomial exact solutions by using lie point symmetry‎. ‎we demonstrate the generation of such polynomial through the medium of the group theoretical properties of the equation‎. ‎a generalized procedure for polynomial solution is pr...

the omega polynomial(x) was recently proposed by diudea, based on the length of stripsin given graph g. the sadhana polynomial has been defined to evaluate the sadhana index ofa molecular graph. the pi polynomial is another molecular descriptor. in this paper wecompute these three polynomials for some infinite classes of nanostructures.

in this paper, we show how certain metabelian groups can be found within polynomial evaluation groupoids. we show that every finite abelian group can beobtained as a polynomial evaluation groupoid.

the relationship between the randic , wiener, hosoya , balaban, schultz indices, harary numbers anddistance matrix to enthalpies of formation (airf), heat capacity, (cp) , enthalpies of combustion (ah °c ),enthalpy of vaporization (ah °vap) and normal boiling points (bpk)of c2 c10 normal alkanes isrepresented

We find an explicit expression of the Tutte polynomial of an $n$-fan. We also find a formula of the Tutte polynomial of an $n$-wheel in terms of the Tutte polynomial of $n$-fans. Finally, we give an alternative expression of the Tutte polynomial of an $n$-wheel and then prove the explicit formula for the Tutte polynomial of an $n$-wheel.

design of crystal-like lattices can be achieved by using some net operations. hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. the networks herein presented are related to the dyck graph and described in terms of omega polynomial and piv polynomials.