# Applied optics and optical engineering,Vol.II by Rudolf Kingslake

By Rudolf Kingslake

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Then: j E l /> n ( n + 1 ) - ( k + l)(n - k)+(n -k), and hence ( E i / > g ( n+ l , k ) . By our hypothesis, D1is strongly Hamilton-connected. Let x1 be a vertex such that (xo,xl) E E. We consider a Hamiltonian path in D1 from xI to xo. It is a Hamiltonian path in D and we can deduce, with the arc ( x o ,xl), a Hamiltonian circuit in D. References [l] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). C. Bermond, A. C. Heydemann and D. Sotteau, Chemins et circuits dans les graphes orient&, Ann.

D, are the degrees of the vertices of a graph G", then (3, N(G",K1,k)=$ i=l for all k 2 2 . Therefore f ( n , K , , k )is just where the maximum is taken over all degree sequences of planar graphs on n vertices. For every n 3 let S" be the graph obtained by joining each of two adjacent vertices to each of the n - 2 vertices of a path of length n - 3. k) = g(n,k ) . k) g(n, k ) . (5) 52 and every (6) We prove (6) for every fixed k by induction on n. If n = 4, (6) is trivial. Assuming it holds for n - 1, let us prove it for n ( n 2 5 ) .

Therefore q 5 2Pl+ 2p3 = 2 p - 4 , again contradicting the hypothesis that G is minimum 2-equi. 3 are greater than zero. Set G ’ = G -{u, W } =(IU u u W), and T = ( U U W), = ( V U W)G. Then since G is 2-equi, d ‘ ( x , y ) S 2 for any x E U and y E W, where d ’ ( x , y ) stands for the distance between x and y in G’ (see Fig. 13). So T lies in a connected component H of G’. O n the other hand, we have: q ( f f ) < q ’ < q -(degu + d e g w ) 2 p - 5 - ( 2 p , + p2 + p 3 ) =p2+p3-1. Miscellaneous properties of equi -eccentric graphs 23 So p ( H ) S p z + p 3 since H is connected.