An algorithm to analyse the polynomial deck of the line graph of a triangle-free graph

An algorithm is presented in which a polynomial deck, 'P D, consisting of m polynomials of degree m-1, is analysed to check whether it is the deck of characteristic polynomials of the one-vertex-deleted sub graphs of the line graph, H, of a triangle-free graph, G. We show that if two necessary conditions on 'P D, identified by counting the edges and triangles in H, are satisfied, then one can construct potential triangle-free root graphs, G, and by comparing the polynomial decks of the line graph of each with 'P D, identify the root graph.peer-reviewe

On the Reconstruction Problem in Graph Theory

The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively. A bi-degreed graph G is a graph with two degrees d &gt; δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d "big" vertex and degree δ "small" vertex. Define "symmetric" path of length p Sp to be one with both ends small vertices and all other internal vertices big vertices; define "asymmetric" path of length p Ap to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is "independent" of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of "forced move" posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also "forced edge" posed by Dr. Swart as well), it's easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two Ss(G)'s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n Ss's following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence Ss's are disjoint. Think of Ss's as "lines" in some geometry. Define two more "distance" functions s1 and s2 such that s1 "represents" the distance from a point to a line and s2 means the distance between to "skew" lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable. A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v1 and v2 such that every edge joins v1 and v2. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G's finiteness, we note they will "terminate" somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition Bi's be that number of special chains is edge-reconstructable(and some more general variations); condition P's be that the "last vertices" of two special chains be not adjacent; we can prove that all A, Bi and P's should hold inductively in an interlocked way. (This is a big task). Then condition P's can be used to prove G's edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.</p

Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory

Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes

Stalin in October

Originally published in 1987. In March 1917 young Joseph Stalin, already a high-ranking Bolshevik, returned from Siberian exile in search of greatness and power. But his activities during the months leading up to the October Revolution were full of blunders and misjudgments—failures that in later years Stalin obliterated from the historical record. Stalin in October reassembles the history of 1917 and explains why, on the eve of the revolutionaries' seizure of power, Stalin seemingly dropped out of the picture. "He would always be dogged," Slusser writes, "by a nagging sense of having somehow missed the revolution." The lingering shame was crucial to Stalin's development into a Soviet dictator

Annual Report of the University, 1992-1993, Volumes 1-4

SIGNIFICANT DEVELOPMENTS Preparation, approval by President Peck, delivery to NMCHE of UNM\u27s response to House Memorials 38 and 25 (on minorities and women). Development and packaging of a presentation on minorities at UNM to Hispanic community people and organizations. Renewal of faculty instructional workload report and other information for use by President Peck and others in the President\u27s Council in testimony to the legislature on accountability by faculty. Significant workload and contributions to WICHE\u27s Diversity Project: - responses to long questionnaire - projected demographics - substitution for O. Forbes on planning for diversity Reprogramming of obsolete computer program of the University of Southern California\u27s Faculty Planning Model. Work remains incomplete. Support and staff work for University Planning Council, Faculty Senate Long Range Planning Committee, Senate President, Senate Budget Committee, Student Learning Outcomes Assessment Committee, Admissions and Registration Committee, Staff Council; Graduate Petition and grade Review Subcommittee Service to NMCHE\u27s Outcomes Assessment Advisory Group; NMCHE\u27s review group on diversity plans Service on Albuquerque Business/Education Compact Conducted several special data analyses to provide user outcome information for the Center for Academic Program Support (CAPS). Wrote reports to summarize analyses. Served in an advisory capacity to VP Zuniga Forbes for the two surveys (Campus Climate for Diversity, ACT Student Opinion Survey) and helped to draw the sample for the ACT survey. Conducted secondary analyses and prepared report of all analyses of the Freshman Survey (CIRP) for VP Zuniga Forbes. Gave presentation of CIRP findings to the Regents Subcommittee on Student Affairs. Conducted secondary analyses and prepared report of all analyses of the Campus Climate for Diversity Survey for VP Zuniga Forbes