Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two "parallel" induced paths. The remaining graphs that guarantee maximum multiplicity 2 are comprised by certain identified families of "exceptional" partial 2-trees that are not linear.
In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than $1$ are simple and also the multiplicity of Laplacian eigenvalue $1$ has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order $n$ that have a multiplicity that is close to the upper bound $\frac{n-3}{2}$, and emphasize the particular role of the algebraic connectivity.
For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which the bound can be decreased.
Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for λ\neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for the multiplicity.The energy of a graph is defined as the sum of absolute values of the eigenvalues. A graph G on n vertices is said to be borderenergetic if its energy equals the energy of the complete graph Kn. We present families of non-cospectral and borderenergetic cographs.
Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $ρ_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $ν(G)$ the independence number of $G$. Recently, the paper [Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, arXiv:1912.13227] discussed the graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$. However, there is one remaining case (graphs with $ρ_{n-1}(G)\neq 1$ and $ν(G)= 2$) not considered. In this paper, we focus on cographs and graphs wi…
Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$. The largest eigenvalue $\partial_{1}^{L}(G)$ is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized …
For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and an upper bound for M(G) respectively, and we show that these bounds are sharp.
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let $ρ_{n-1}(G)$ and $ν(G)$ be the second least normalized Laplacian eigenvalue and the independence number of a graph $G$, respectively. As the main conclusions, two families of $n$-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$ are determined: graphs with $ρ_{n-1}(G)=-1$ and graphs with $ρ…
Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one of the Laplacian eigenvalue having multiplicity $k$. It is well known that $\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined by Das, and the graphs of $\mathcal{G}(n,n-3)$ with four distinct Laplacian eigenvalues are determined by Mohammadian et al. In this paper, we determine the graphs of $\mathcal{G}(n,n-3)$ with three distinct Laplacian eigenvalues, and then the full characterization of the graphs in $\mathcal{G}(n,n-3)$ is completed.
For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\m…
Let $G$ be a connected simple graph of order $n$. Let $ρ_1(G)\geq ρ_2(G)\geq \cdots \geq ρ_{n-1}(G)> ρ_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(ρ_i)$ the multiplicity of the normalized Laplacian eigenvalue $ρ_i$. Let $ν(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365…
Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $ρ$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by $L_{A}^ρ$ the generalized Laplacian matrix $D(A)-ρA$. A well-known result of Grone et al. states that by connecting one of the end-vertices of $P_{3}$ to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue $1$. We extend this theorem and some other results for a given generalized Laplacian eigenvalue $μ$. Furthermore, we giv…
Given a graph $G$, one may ask: "What sets of eigenvalues are possible over all weighted adjacency matrices of $G$?" (The weight of an edge is positive or negative, while the diagonal entries can be any real numbers.) This is known as the Inverse Eigenvalue Problem for graphs (IEP-$G$). A mild relaxation of this question considers the multiplicity list instead of the exact eigenvalues themselves. That is, given a graph $G$ on $n$ vertices and an ordered partition $\mathbf{m}= (m_1, \ldots, m_\ell)$ of $n$, is there a weighted adjacency matrix where the $i$-th distinct eigenvalue has multiplici…
The Laplacian matrix of a graph $G$ is denoted by $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two graphs. A one-edge connection of two graphs $G_1$ and $G_2$ is a graph $G=G_1\odot_{uv} G_2$ with $V(G)=V(G_1)\cup V(G_2)$ and $E(G)= E(G_1)\cup E(G_2)\cup \{e=uv\}$, where $u\in V(G_1)$ and $v\in V(G_2)$. We investigate the multiplicity of the Laplacian eigenvalue $2$ of $G_1\odot_{uv} G_2$, while the unicyclic graphs $G_1$ and $G_2$ have $2$ among their Laplacian eigenvalues, by using their…
Let $ Φ=(G, φ) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ Δ$. The associated adjacency matrix and degree matrix are denoted by $ A(Φ) $ and $ D(Φ) $, respectively. Let $ m_α(Φ,λ) $ be the multiplicity of $ λ$ as an eigenvalue of $ A_α(Φ) :=αD(Φ)+(1-α)A(Φ)$, for $ α\in[0,1) $. In this article, we establish that $ m_α(Φ, λ)\leq \frac{(Δ-2)n+2}{Δ-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(Φ)$ in terms of the…
Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree $n$-vertex graph? The best known upper bound is $O(n/\log\log n)$. The previously known best known lower bound is on the order of $n^{1/3}$ (for infinitely many $n$), coming from Cayley graphs on $\text{PSL}(2,q…
We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.
Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $θ$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise angles $θ$ exceeds $n+Ω(\log\log n)$ but is at most $n+o(n)$. To accomplish this, we construct, for various real $λ$ and integer $d$, $d$-regular graphs with second eigenvalue exactly $λ$ and arbitrarily large second eigenvalue multiplicity. Central to our construction is a distribution on factors of bipartite graphs which possesses concentration properties.
Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characteri…
An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the Möbius-Kantor graph, the Nauru graph, and the Foster g…
We study various spectral properties of the Seidel matrix $S$ of a connected chain graph. We prove that $-1$ is always an eigenvalue of $S$ and all other eigenvalues of $S$ can have multiplicity at most two. We obtain the multiplicity of the Seidel eigenvalue $-1$, minimum number of distinct eigenvalues, eigenvalue bounds, characteristic polynomial, lower and upper bounds of Seidel energy of a chain graph. It is also shown that the energy bounds obtained here work better than the bounds conjectured by Haemers. We also obtain the minimal Seidel energy for some special chain graphs of order $n$.…
The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix with the SSP (or the SMP) then a supergraph has a matrix with the same spectrum (or ordered multiplicity list) augmented with simple eigenvalues if necessary, that is, subgraph monotonicity. In this paper we extend this to a form of minor monotonici…
If $G$ is a graph and $\mathbf{m}$ is an ordered multiplicity list which is realizable by at least one symmetric matrix with graph $G$, what can we say about the eigenvalues of all such realizing matrices for $\mathbf{m}$? It has sometimes been tempting to expect, especially in the case that $G$ is a tree, that any spacing of the multiple eigenvalues should be realizable. In 2004, however, F. Barioli and S. Fallat produced the first counterexample: a tree on 16 vertices and an ordered multiplicity list for which every realizing set of eigenvalues obeys a nontrivial linear constraint. We extend…
For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $τ(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $τ(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $λ\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $λ\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show tha…
Among those real symmetric matrices whose graph is a given tree $T$, the maximum multiplicity $M(T)$ that can be attained by an eigenvalue is known to be the path cover number of $T$. We say that a tree is $k$-NIM if, whenever an eigenvalue attains a multiplicity of $k-1$ less than the maximum multiplicity, all other multiplicities are $1$. $1$-NIM trees are known as NIM trees, and a characterization for NIM trees is already known. Here we provide a graph-theoretic characterization for $k$-NIM trees for each $k\geq 1$, as well as count them. It follows from the characterization that $k$-NIM tr…
In this paper, we completely characterize the graphs with third largest distance eigenvalue at most $-1$ and smallest distance eigenvalue at least $-3$. In particular, we determine all graphs whose distance matrices have exactly two eigenvalues (counting multiplicity) different from $-1$ and $-3$. It turns out that such graphs consist of three infinite classes, and all of them are determined by their distance spectra. We also show that the friendship graph is determined by its distance spectrum.
A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an eigenvalue in the interval $(-1,0)$. It is also shown that the multiplicity of any eigenvalue of a cograph $G$ does not exceed the sum of multiplicities of $0$ and $-1$ as eigenvalues of $G$. We introduce a partial order on the vertex set of graphs $G$ in terms of inclusions among the open and closed neighborhoods of vertices, and conjecture that the multipl…
This article focuses on finding the eigenvalues of the Laplacian matrix of the comaximal graph $Γ(\mathbb Z_n)$ of the ring $\mathbb Z_n$ for $n> 2$. We determine the eigenvalues of $Γ(\mathbb Z_n)$ for various $n$ and also provide a procedure to find the eigenvalues of $Γ(\mathbb Z_n)$ for any $n> 2$. We show that $Γ(\mathbb Z_n)$ is Laplacian Integral for $n=p^αq^β$ where $p,q$ are primes and $α, β$ are non-negative integers. The algebraic and vertex connectivity of $Γ(\mathbb Z_n)$ have been shown to be equal for all $n> 2$. An upper bound on the second largest eigenvalue of $Γ(\mathbb Z_n)…
Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the $n$-Queens' graph $\mathcal{Q}(n)$ is equal to $-4$ for every $n \ge 4$ and it is also proven that the multiplicity of this eigenvalue is $(n-3)^2$. Additionally, some results on the edge clique partition graph parameters are obtained.
We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity $1$, and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues, etc.
Let $T$ be a tree with a given adjacency eigenvalue $λ$. In this paper, by using the $λ$-minimal trees, we determine the structure of trees with a given multiplicity of the eigenvalue $λ$. Furthermore, we consider the relationship between the structure of trees and the eigensystem of a given Laplacian eigenvalue.
We show that every $0$-$1$ multiplicity matrix for a simple graph $G$ is generically realisable for $G$. In particular, every multiplicity matrix for a path is generically realisable. We use this result to provide several families of joins of graphs that are realisable by a matrix with only two distinct eigenvalues.
Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers critical groups over polynomial rings.
