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If I understood correctly, you do not have any information about $S$. In general, the answer to your question is no. The group $Z_2^n$ is not $Cay-DS$ in general. So, for suitable $n$, you can find tw …

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 …

It is a partial answer for your question: For $P_n$, the form of normalized laplacian matrix is three diagonal and with some calculations, we can show that all its normalized laplacian eigenvalues ar …

I think this question is so hard, since we do not have any control on other eigenvalues, specially on the minimum of them. As an evidence (and maybe useful for your work), recently S. M. Cioab‎$‎\br …

We know that $DS$ graphs are such connected graphs that determinable by their adjacency spectrum. Suppose $DS(n)$ and $G(n)$ show the number of $DS$ graphs and all graphs with $n$ vertices,respective …

Let $\mathcal{G}$ be the set of all finite connected simple graphs minus the complete graphs. For any $G\in \mathcal{G}$, let $\lambda_{\geq0}(G)$ denotes the smallest positive adjacency eigenvalue of …

We have the spectrum and the degree sequence of one graph. Can we uniquely determine the graph with these given information?

There are some facts that can be found by the spectrum of adjacency matrix of graph.For example, the number of edges and vertices, is bipartite or not, is complete multipartite or not and so on. Can w …

Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that are obtained by applying a binary operation to $G$ and $H$? For example, to take one s …

Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that: $1)$ $G_i‎\ncong H_i$ for $i=1, 2, \cdots, n$ $2)$ …

Let $G$ be a simple graph. We say this graph is self-index complementary ($SIC$) if $\lambda_1 (G)=\lambda_1 (\overline{G})$, where $\lambda_1(G)$ denotes the index of the adjacency matrix of the grap …

As @Morris answered the reason is behind in connectivity and rapid connection which is compacted in isoperimetric parameter of graphs. The isoperimetric parameter has bounded by eigenvalues in some ni …

The simple connected graph $G$ has $n$ vertices and we have: 1) $|E(G)|‎\geq‎ \frac{n(n-1)}{3}$ 2) we have the spectrum and degree sequence of $G$ 3) $Spectrum(G)=Spectrum(H)$ Is $G \cong ‎H$?

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas. In the following all graphs are simple and connected. Let $G$ be graph with vertex set $V=\left\{v_1,v_2,\ld …

I am looking for a large family (infinite pairs) of cospectral graphs with these condtions: The graphs are non-regular, Minimum degree is greater than $1$, The degree sequences of these cospectral …