In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and Bose proved that every translation plane coordinatized by a quasifield that is finite-dimensiona...

In 1982, B. Bollobas and Vega in the paper gave the configurational model to generate $r$-regular random graphs. They gave the following theorem (Theorem 1 in the paper). Theorem: Let $r\geq 3$ and $\epsilon> 0$ be fixed and define $d=d(n)$ as the least integer satisfying $$(r-1)^{d-1}\geq (2+\e...

Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular. Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both? Thanks in advance.

Let $G$ be a regular simple graph with degree $\Delta=n-k-1$ and order $m$. Let $C_k$ be the regular graph which is formed by removing a $k$-factor from the complete graph $K_{n}$. I think we could always find a proper induced subgraph of $C_k$ with maximum degree at least $\ge\frac{\Delta}{2}$ ...

There exist inequalities relating the maximum and minimum eigenvalues of the adjacency matrix of a graph with its chromatic numbers, i.e. the Wilf's and Hoffmann's inequalities, which put together state that $1-\frac{\lambda_{max}}{\lambda_{min}}\le\chi(G)\le 1+\lambda_{max}$, where $\chi, \lambd...

There is a related question on how to generate all regular graphs; however, the procedure is random and repeats the generated graphs. Plus, there is no stop condition, unless recording the total number of non-isomorphic graphs and matching it to the known numbers. I wonder if there is a more effi...

What is the smallest 3-regular graph to have a unique perfect matching? With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in this question Cubic graphs without a perfect matching and a vertex incident to three bridges ). So ...

In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and Bose proved that every translation plane coordinatized by a quasifield that is finite-dimensiona...

Hey Everyone! In nearly all (if not all) projective geometry texts I have bumped into the following theorem: "Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word line and the corresponding incidence relations once again one obtains a theorem of $\math...