Let R denote the family of functions f(z)=z+∑n=2∞anzn of bounded boundary rotation so that Ref′(z)>0 in the open unit disk U={z:z<1}. We obtain sharp bounds for Toeplitz determinants whose elements are the coefficients of functions f∈R.

We consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

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Let G be a simple connected graph of order n. Let D c ν e (G, i) be the family of connected vertex-edge dominating sets of G with cardinality i. The polynomial D c ν e (G, x) = D c υ e ( G x ) = ∑ ​ i = γ c υ e n ( G ) d c υ e ( G i ) x i is called the connected vertex-edge domination polynomial of G where d c ν e (G, i) is the number of vertex edge dominating sets of G. In this paper, we study some properties of connected vertex-edge domination polynomials of the Gem graph G n . Also we obtain some properties of D c ν e (G n , x) and their coefficients. Also, we find the recursive formula, to derive the connected vertex-edge dominating sets of the Gem graph G n .