On Murty-Simon Conjecture II

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$. In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al. is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. In this paper, we completely prove the Murty-Simon Conjecture for the graphs whose complements have vertex connectivity $\ell$, where $\ell = 1, 2, 3$; and for the graphs whose complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201

Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings

AUTOMATED CONJECTURING ON THE INDEPENDENCE NUMBER AND MINIMUM DEGREE OF DIAMETER-2-CRITICAL GRAPHS

A diameter-2-critical (D2C) graph is a graph with diameter two such that removing any edge increases the diameter or disconnects the graph. In this paper, we look at other lesser-studied properties of D2C graphs, focusing mainly on their independence number and minimum degree. We show that there exist D2C graphs with minimum degree strictly larger than their independence number, and that this gap can be arbitrarily large. We also exhibit D2C graphs with maximum number of common neighbors strictly greater than their independence number, and that this gap can be arbitrarily large. Furthermore, we exhibit a D2C graph whose number of distinct degrees in its degree sequence is strictly greater than its independence number. Additionally, we characterize D2C graphs with independence number 2 and show that all such graphs have independence number greater or equal to their minimum degree