JOURNAL OF GRAPH THEORY, VOL. I, 11-12 (1977)

A Note of Welcome

ED WARD WRIGHT University of Aberdeen

Over the door of the church at Staunton Harold, the only church built in England during Cromwell’s dictatorship, is the inscription: “In the year 1653 when all things Sacred were throughout ye nation Either de- molished or profaned Sir Robert Shirley, Barronet, Founded this church; whose singluar praise it is to have done ye best things in ye worst times And hoped them in the most calamitous.” Things are not quite as bad as that at the moment but it is a pleasure to salute the courage of those starting a new journal at the present time when finance for scientific research and university libraries is restricted and the cost of producing a journal has increased so much. However, the fascinating subject of Graph Theory, to which the new journal is devoted, is certainly itself flourishing and expanding.

While I had always been interested in the combinatorial aspects of Number Theory, my attention was first drawn to Graph Theory as a subject for research by Gilbert’s relationship between the generating functions for the numbers of labeled graphs on n labeled nodes and of those which are connected [l]. (One function is the formal logarithm of the other.) The series for the generating functions diverge for all values of the variable except zero and so I could not apply Cauchy’s Theorem, my usual weapon. However, the asymptotic relationship between two such sequences intrigued me and I wrote two papers on the subject [2, 31. I was so fortunate as to have these reviewed for Mathematical Reviews by Professor R. C. Read who wrote to me, explained the graph theory significance of my results and asked me if I could apply my method to the corresponding (but different) relationship for graphs with ,unlabeled nodes. I could do this and so was led to my asymptotic results for the number of unlabeled graphs on n nodes and q edges [4], and to a number of further theorems and problems.

Hardy and Littlewood’s Partitio Numerorum papers in the twenties contained a partial solution of Waring’s Problem for large enough inte- gers. There were subsequent improvements by Vinogradoff and others. In the thirties L. E. Dickson of Chicago and S. S . Pillai of Bombay found a solution for all integers and all but a few kth powers by heavy computa- tion based on the asymptotic work. I was particularly aware of this, since @ 1977 by John Wiley & Sans, Inc. 11

12 WRIGHT

for some time Dickson and Pillai were unaware of each other’s work and all their immensely complicated papers were sent to me by the Zentralb- latt to abstract. As a consequence I have always hoped to use some of my asymptotic work in Graph Theory to get exact results for all n. Comput- ing seems unlikely to close the gap between small and large n, owing to the combinatorial explosion, but the methods used to soIve the asympto- tic problems have enabled me to solve two problems for all n, one a simplification of Liskovec’s method to calculate the number of strongly connected digraphs [5] and the other a method of calculating a formula for the number of labeled connected graphs on n nodes and n + k edges [ 6 ] .

Throughout the whole of the period during which I have been working on Graph Theory, I have been primarily engaged in administration. It has been a pleasure and a relief to retreat occasionally from the mild irrationality of academic politics to the rationality of graph theory, though both are very interesting in their different ways. I am looking forward to my retirement from administration this year to devote my whole time to mathematical research and I hope to inflict an occasional paper on the editors of this new journal.

When I had finished writing this, I attended a meeting of my University Library Committee, a t which we were considering curtailing our list of scientific periodicals for financial reasons and certainly not taking any new ones. However, this periodical will be an exception to the genera1 rule.

REFERENCES

1 . E. N. Gilbert, Enumeration of labeled graphs. Cun. J. Math. 8 (1956) 405411. 2. E. M. Wright, A relationship between two sequences. Proc. London. Math. Soc. 17

3. E. M. Wright, A relationship between two sequences. 11. Proc. London. Math. Soc. 17

4. E. M. Wright, Graphs on unlabeled nodes with a given number of edges. Acfa Math. 126

5 . E. M. Wright, The number of strong digraphs. Bull. London Math Soc. 3 (1971)

6 . E. M. Wright, The number of connected (n , n + k ) graphs. Notices Amer. Math. Soc. 23

(1967) 296-304.

(1967) 547-552.

(1971) 1-9.

348-350.

(1976) A-5.