An answer to a question of Isaacs on character degree graphs

Character Degree Graphs That Are Complete Graphs

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) − {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

When to Reproduce? A New Answer to an Old Question.

We present a life-history model based on the assumptions that juvenile survival follows a negative exponential function and that fecundity gain increases linearly with time to maturity. This model predicts that the optimal fitness is achieved when survival at maturity is 0.368 (e(-1)). Survival at the time of maturity is therefore an invariant. We tested this prediction by using published data ...

On reverse degree distance of unicyclic graphs

The reverse degree distance of a connected graph $G$ is defined in discrete mathematical chemistry as [ r (G)=2(n-1)md-sum_{uin V(G)}d_G(u)D_G(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $G$, respectively, $d_G(u)$ is the degree of vertex $u$, $D_G(u)$ is the sum of distance between vertex $u$ and all other vertices of $G$, and $V(G)$ is the...

On Foguel's Answer to Nagy's Question

Nagy's question is whether or not every power-bounded operator is similar to a contraction [3]. ("Power-bounded" means that the norms of the positive powers are bounded.) Foguel's answer is no [l]. The purpose of this note is to look at Foguel's ingenious counterexample from a point of view somewhat different from his own. The advantage of the new look is that it is less computational; its draw...

A Question and Answer Session on the Meaning of Being Human , with Dr. Gholamreza Aavani