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This conjecture is not true in general. For example, let $G$ be a graph that obtained from joining the end vertex of $P_3$, $P_3$ and $P_4$, where it is an star-like tree. This graph has largest eigen …

Let $P_n$ be a permutation of $1,2,\ldots,n$. Also, suppose $is(P_n)$ and $ds(P_n)$ shows the longest increasing and decreasing subsequences of permutation $P_n$, respectively. So, the Erdős–Szekeres …

Suppose $G$ is a finite group and we choose $S'=G\setminus\{1,a,a^{-1}\}$. Then if we let $S=S'\cup \{a,a^{-1}\}$, the diameter of the cayley graph $Cay(G,S)$ is one, but the diameter of the Cayley gr …

This problem is so famous. For first trivial reference, you can see:link. $\it{R. Bodendiek}$ and $\it{G. Burosch}$ studied this problem in a paper with name: "Solution to the Antimagic 0,1,-1 Matri …

I think you mean the adjacency spectrum of simple graph. Since the eigenvalues of graphs are algebraic integers, so the answer to your question is no. For example, the number $\frac{226}{17}$, which i …

There are some results about $P_n$-free graphs. For example; A graph $G$ is $P_4$-free if and only if each connected induced subgraph of $G$ contains a dominating induced $C_4$ or a dominating vertex …

I write this as an answer since I need some vote to break the symmetry of my reputation. I hope I never fall down to the other symmetry. The answer of your question is yes. Actually you can see the P …

We know that all groups with prime order is CI-group. So, if two circulant graph with $p$ vertices be cospectral, then they are isomorphic. So, as Dear Brendan said, the answer is no. But about your …

You can find your answer in the paper: Generalized Girth Problems in Graphs and Hypergraphs By Uriel Feige and Tal Wagner. Let $A_2(n,k)$ denotes the optimal upper bound for girth of graphs with $n$ n …

Do we have any formula for counting the number of graphs with $n$ vertices, that has exactly $k$ vertices with degree $d$ and the other vertices have different and disjoint degrees? (Different and dis …

There are a lot of properties which $G$ and $\overline{G}$ both have them: number of vertices (edges), automorphism group, and etc. So, naturally this question is interesting for me. I want to mention …

For Abelian group $G$, the eigenvalues of the Cayley graph $Cay(G,S)$ can be computed by the group characters and the set $S$. Actually, if $\rho$ be a character of the group $G$, $\rho(S)=\Sigma_{s\i …

Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$. I need some groups such t …

Since my comment is long, I write it as an answer, but it is not a complete answer and just give some insight. Firstly, based on the paper you mentioned and based on the applications of MDS matrices …

You are right Dear domotorp, Suppose the second largest eigenvalue of bipartite graph $G$ is one, i.e, $\lambda_2=1$. In this case, $G$ belong to the finite type, (there are infinite number of such g …