Solvable groups whose character degree graphs generalize squares

Let G be a solvable group, and let {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.

Learning socio-organizational network structure in buildings with ambient sensing data

We develop a model that successfully learns social and organizational human network structure using ambient sensing data from distributed plug load energy sensors in commercial buildings. A key goal for the design and operation of commercial buildings is to support the success of organizations within them. In modern workspaces, a particularly important goal is collaboration, which relies on physical interactions among individuals. Learning the true socio-organizational relational ties among workers can therefore help managers of buildings and organizations make decisions that improve collaboration. In this paper, we introduce the Interaction Model, a method for inferring human network structure that leverages data from distributed plug load energy sensors. In a case study, we benchmark our method against network data obtained through a survey and compare its performance to other data-driven tools. We find that unlike previous methods, our method infers a network that is correlated with the survey network to a statistically significant degree (graph correlation of 0.46, significant at the 0.01 confidence level). We additionally find that our method requires only 10 weeks of sensing data, enabling dynamic network measurement. Learning human network structure through data-driven means can enable the design and operation of spaces that encourage, rather than inhibit, the success of organizations.

ON THE REGULARITY OF CHARACTER DEGREE GRAPHS

Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$. The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$-regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.