Related papers: Spectrum of the Transposition graph

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an…

We study the derangement graph $\Gamma_n$ whose vertex set consists of all permutations of $\{1,\ldots,n\}$, where two vertices are adjacent if and only if their corresponding permutations differ at every position. It is well-known that…

We study the adjacency spectrum of the toroidal three-dimensional queen graph $G_n$ on $(\mathbb{Z}_n)^3$. Since $G_n$ is a Cayley graph on an abelian group, its adjacency matrix is diagonalized by Fourier characters. For each frequency…

For $n\geq 3$, define $T_n$ to be the theory of the generic $K_n$-free graph, where $K_n$ is the complete graph on $n$ vertices. We prove a graph theoretic characterization of dividing in $T_n$, and use it to show that forking and dividing…

Threshold graphs are graphs that can be characterized in a number of different ways. For example, they are graphs that are $P_4,\ C_4,\ 2K_2$--free. They may also be characterized by a finite sequence of positive integers $a_1, \ldots,…

Let $\Gamma$ be a simple graph with $n$ vertices. The energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The graph $\Gamma$ is said to be hyperenergetic if…

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever…

The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the…

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest…

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we show a cograph that has a balanced cotree $T_{G}(a_{1},\ldots,a_{r-1},0|0,\ldots,0,a_{r})$ is integral computing its spectrum. As an…

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations…

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$…

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum.

A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy…

Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1.…

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated…

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible…

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…

The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread…

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.