The terminal Hosoya polynomial of a graphGis defined asTH(G,λ)=∑k≥1‍dT(G,k)λk, wheredT(G,k)is the number of pairs of pendant vertices ofGthat are at distancek. In this paper we obtain terminal Hosoya polynomial of line graphs.

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where |V | = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i = j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β o , μ)-designs over the graphs K n × K n , T (n)-triangular graphs, L 2 (n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β o , μ)-designs over the three Chang graphs T 1 (8), T 2 (8) and T 3 (8).

For any graph G(V,  E), a function f : V (G)    0, 1, 2, 3     is called Double Roman dominating function (DRDF) if the following properties holds, If f (v) = 0, then there exist two vertices v1, v2 ∈ N (v) for which f (v1) = f (v2) = 2 or there exist one vertex u ∈ N (v) for which f (u) = 3.∈ If f (v) = 1, then there exist one vertex u N (v) for which f (u) = 2 or Σ f (u) = 3. The weight of DRDF is the value w(f ) = v∈V (G) f (v). The minimum weight among all double Roman dominating function is called double Roman domination number and is denoted by γdR(G). In this article we initiated research on double Roman domination number for middle graphs. We established lower and upper bounds and also we characterize the double Roman domination number of middle graphs. Later we calculated numerical value of double Roman domination number of middle graph of path, cycle, star, double star and friendship graphs.

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