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Your irregular $(3,2)$-nice graph is almost a De Bruijn graph. Label the vertices ($0$ to $8$) with $12,21,11,22,10,02,20,01,00.$ Then the deviations are that your edges $$1\rightarrow 6 \ \&\ 3\righ …

For $v=6k+1$ or $v=6k+3$ and $k$ not extremely small there are a great many ways to make a Steiner Triple System with $v$ objects and $M=\frac{v(v-1)}6$ triples so that each pair of objects is in a un …

It has nothing essential to do with polynomials. Just sets of numbers. Let $\beta_{11} \lt \beta_{21} \lt \cdots \beta_{n1}$ and $\beta_{12} \lt \beta_{22} \lt \cdots \beta_{n2}$ be two lists. They h …

Here is a ridiculous solution using a $1 \times 1$ matrix! First, encode an $n \times n$ adjacency matrix $A$ by $a \lt 2^{\binom{n}2}$ where $a$ is the binary integer obtained by listing the above di …

The lower bound is less than $$\frac{\deg_{\max}}2+1$$ but not by much. Here is (part of) a graph with an edge adjacent to $10$ others and (part of) the corresponding line graph. The largest eigen …

As I mentioned in a comment. Two $n$-vertex graphs , both regular of degree $d$ , will have this property, although the graphs can be quite different in some ways, for example one may be connected and …

I'm not sure how common this is. It may depend on how you are determining random graphs. There may be ties so let me separate the issues of the equivalence relation "same rank" from those of the lin …

Here are a few ideas on places to look for examples. You may not find (m)any this way. First to eliminate a trivial case: For some people a complete graph or disjoint union of isomorphic complete gra …

As i commented, you need to be more specific about the details. Here is an idea though which might work well if within groups there are lots (or at least a reasonable number of) triangles but no trian …

It is interesting, worthwhile, and not very difficult to understand the situation for the Cartesian product of two or more arbitrary graphs. However here is a simplified answer which applies to this c …

Here is something which seems to work. Consider a circulant graph with vertices $v_0,\cdots,v_{2^n-1}$ and $v_i$ adjacent to $v_{i\pm d}$ where $d$ ranges over a set $D$ of $\lceil\frac{n+1}{2}\rceil …

The question is fairly broad. I can say a number of things, but none of them are very deep. For a regular graph the largest eigenvalue of the adjacency matrix, $A$, is the common degree with eigenvect …

Look around. The term cospectral is also used. Some of the people who have been answering you showed that Almost all Trees are Co-spectral. Of course trees can be quite irregular. An early paper is Co …

I assume you mean to fix $d$ and let $n$ grow. When $d \gt 3$ the graph (at least in the case that all vertices have degree $r$) will likely be connected and even $d$-connected. But in case $d=2$ one …

Maximum degree 2 would be such a class (which includes regular of degree $2$ as a subclass). Transitive graphs (by which I mean that the relation of being connected by an edge is transitive) are anoth …