Halving complete 4-partite graphs

We completely determine the spectrum (i.e. set of orders) of complete 4-partite graphs with at most one odd part which are decomposable into two isomorphic factors with a finite diameter. For complete 4-partite graphs with all parts odd we solve the spectrum problem completely for factors with diameter 5. As regards the remaining possible finite diameters, 2,3,4, we present partial results, focusing on decompositions of K-n,K-n,K-n,K-m and K-n,K-n,K-m,K-m for odd m and n

Practicability of Audit Expert

In this paper we consider the applicability of an auditing method for the prevention of statistical database compromise through data inference. In 1982 Chin and Oszoyoglu proposed Audit Expert for the prevention of such a compromise using SUM queries. As originally proposed, Audit Expert was meant to be used dynamically and it was considered suitable only for small databases. In the static mode Audit Expert is much more efficient both in the storage and time required and so can be used for large databases as well as small ones. Moreover, in static mode it is possible to achieve maximum usability of the database. However, sometimes this will prevent the system from answering queries which are particularly important. In this paper we consider the inclusion of a limited number of user-posed queries and its impact on the usability. Then we say that the Audit Expert is used in a hybrid mode. We prove that maximising usability in a general hybrid Audit Expert is NP-complete and we isolate ce..

Graphs, 0-1 Matrices, and Usability of Statistical Databases

The motivation for this work comes from a security problem of statistical databases: In a database of n records, given k SUM queries, is it possible to answer all of them, plus another \Gamma n b n 2 c \Delta \Gamma k distinct SUM queries, in such a way that no individual value from the database is revealed? The corresponding mathematical problem (stated in terms of certain extensions of 0-1 matrices) is known to be NP-complete in general. We show that it remains NP-complete even when restricted to the case when each query involves four records and each record is in at most three queries. On the other hand, we identify certain cases in which the problem is solvable in polynomial time. The case when every record is contained in at most two of the given k queries is studied in detail from the graph-theoretic point of view. 1 Introduction In its abstract and simplified model, a database can be viewed as a two-dimensional table whose rows are called records, and columns are called ..

The spectra of lifted digraphs

We present a method to derive the complete spectrum of the lift Γα of a base digraph Γ , with voltage assignment α on a (finite) group G. The method is based on assigning to Γ a quotient-like matrix whose entries are elements of the group algebra C[G] , which fully represents Γα . This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs

Halin's Theorem for Cubic Graphs on an Annulus

Halin's Theorem characterizes those locally-finite, infinite graphs that embed in the plane without accumulation points by giving a set of six topologically excluded subgraphs. We prove an analogous theorem for cubic graphs that embed in an open annulus without accumulation points, finding the complete set of 29 excluded subgraphs

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertic..