FROM PATHS TO STARS

The number of cycles in the complement T′ of a tree T is known to increase 
with the diameter of the tree. A similar question is raised and settled for the number of 
complete subgraphs in T′ for a special class of trees via Fibonacci numbers. A structural 
characterization of extremal trees is also presented.

The problem of characterizing the n-vertex trees T for extremal values with respect to c(T') or i(T') loses some structural significance in the generality of Y. Suppose we consider a class of trees which keep out all paths and stars, for example, the class .T'3 of all those trees T3 having exactly three endvertices.What structural similarities between c(T) and i(T;) are inherited from c(T') and i(T')?We recall that the diameter of a graph G is the maximum distance d(u, v) taken over all pairs of vertices u, v in G.The following theorem [1, p.93] relates c(T') with the diameter of T. THEOREM 1.For each n >_ 6 and every tree T of order n and diameter d, 4 _< d _< n 2, there is a tree T1 of order n and at least diameter d + such that c(T') < c(T).
with the largest number of cycles in its complement is a tree with the largest diameter as shown in Figure 2.

T Figure 2. A tree with maximum c(T).
So, among all trees with three cndvcrtices, of order n, the direct relationship between c(T) and the diameter of Ta is inherited from the class of all trees, i.e., rain c(T) and max c(T) are still associated with the smallest and the largest diameters of T3, respectively.
Can wc make the same claim about i(T), the number of complete subgraphs in the comple- ment of T37 We recall that when T is arbitrary, i(T') is maximum when the diameter of T is minimum (T is a star) and i(T') is minimum when the diameter is maximum (T is a path).Does this relationship between i(T') and the diameter of T remain true when T is restricted to 37 To this end, we need the concept of a Fibonacci number .f(G) of a graph G.

MAIN RESULTS
If T3 is a tree with three endvertices, then it has a unique vertex u of degree three.We count i(T) [5] by considering two disjoint sets of complete subgraphs of T, say S and $2, where S is the set of those complete subgraphs not containing the vertex u, and 5' consists of those that do contain u.Let vx, v and v3 be the three vertices adjacent to u in T3.Wc i(T) IS, + ISl-i(T , + i(T ,,, ,, ,,)'.
If n 3k + 1, T u is a union of three disjoint paths on 3k vertices, where T v v v is also a union of three disjoit patls on 3k-3 vertices together with the isolated vertex u (see Figure 1).The following theorem on Fibonacci numbers shows that i(T)is minimized and maximized by the trees in Figures 3a and 3b, respectively.This shows (a) min i(T) (b) max i(T) Figure 3. Extremal trees in .'a.
that the inverse relationship between i(T') and the diameter of T is not inherited in the class of trees ."awith exactly three endvertices.
THEOREM 2. Let n be an integer >_ 7. Then among all summands rl,s, and satisfying (i) r --ql + 1 t + 2, (ii) r rz + 1,Sl a2 -I-1,tx tz + 1, and rl,S,tl _ 2 we have the sum of the two products Fr F, Ft + FrF, zFtz of the Fibonacci numbers Fx, F,, Ft and Frz, Fz, Ftz is (a) minimum if ra s 3 and t n -4 and (b) maximum if va s 2 and tl n-2.
That is, min i(T) is realized in Figure 3a.
ACKNOWLEDGEMENTS.The author gratefully acknowledges the support of King Fahd University of Petroleum and Minerals and wishes to thank the anonymous referee for his valuable hints.