Tag Archives: power graph

Graphs on groups, 8

The dark clouds seem to have lifted a bit. Perhaps now, that the last rush of conferences for a while is over, life can return to something like normality … For me the most significant event was the last in … Continue reading

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A little problem

In connection with the power graphs of unitary groups, I came across the following little number-theoretic conundrum. Can anyone solve it? Let q be an odd power of 2 (bigger than 2). Show that (q2−q+1)/3 is not a prime power … Continue reading

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A new constant?

This is an appeal for help. Has anyone come across the constant 2.648102…? Here is the background, which connects with my previous posts about graphs on groups. We are interested in the clique number of the power graph of the … Continue reading

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Graphs on groups, 5

I gave two lectures on this stuff to a new research seminar on Groups and Graphs, run by Vijayakumar Ambat in Kochi, Kerala. The first was an introduction to the hierarchy, the second was about cographs and twin reduction, why … Continue reading

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Graphs on groups, 4

Here is a small problem, mixing group theory and number theory, which might appeal to someone. A couple of definitions. The power graph of a group G has an edge from x to y if one is a power of … Continue reading

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Graphs defined on groups

I’ve been spending time I can ill afford on various questions related to graphs whose vertex set is a group G and which capture some of the structure of G. (I interpret this to mean that the graph is invariant … Continue reading

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Induced subgraphs of power and commuting graphs

For those who like thinking about these things, here is a small observation and a few problems. As I have recently discussed, the power graph of a group is perfect. This means that all its induced subgraphs are perfect, and … Continue reading

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Graphs defined on groups

Apologies; I have been so busy lately that very little has got written up. Let me try to remedy this with a quick tour through some recent mathematical developments. As some of my posts have hinted, one topic I have … Continue reading

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Perfectness of the power graph

The power graph of a group is the graph whose vertices are the group elements (sometimes the identity is excluded but it doesn’t matter here), in which x and y are joined if one is a power of the other. … Continue reading

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Groups and graphs

Two new preprints have appeared on the arXiv (2008.09291 and 2009.02884), about which I would like to say something. One is by Saul Freedman, the other by him and two co-authors (his PhD supervisors). The intersection graph of a group … Continue reading

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