Related papers: Median eigenvalues of bipartite subcubic graphs

Given a graph $G$, let $\lambda_3$ denote the third largest eigenvalue of its adjacency matrix. In this paper, we prove various results towards the conjecture that $\lambda_3(G) \le \frac{|V(G)|}{3}$, motivated by a question of Nikiforov.…

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to…

Suppose $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d_G( v_i,v_j ) $ be the least distance between $v_i$ and $v_j$ in $G$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} ) _{n\times…

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.

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.

Given a graph $H$, we investigate the $d$-regular graphs $G$ with the highest $H$-density. We reframe the problem as a continuous optimization problem on the eigenvalues of $G$ by relating injective homomorphism numbers from $H$ and…

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le…

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that…

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order…

Let $G$ be a unicyclic graph. In this paper, we provide an upper bound for the number of Laplacian eigenvalues of $G$ within the interval $[0,1)$ in terms of the diameter and the girth of $G$.

Let $\eta_{1}\ge \eta_{2}\ge\cdots\ge \eta_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $\eta_{n}>-1$. As a result, we determine all connected graphs…

We determine all Hermitian $\mathcal{O}_{\Q(\sqrt{d})}$-matrices for which every eigenvalue is in the interval [-2,2], for each d in {-2,-7,-11,-15\}. To do so, we generalise charged signed graphs to $\mathcal{L}$-graphs for appropriate…

The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times…

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications,…

The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any…

A pseudo $(v,\, k,\, \la)$-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in $\la$ elements; and…

A neutral network is a subgraph of a Hamming graph, and its principal eigenvalue determines its robustness: the ability of a population evolving on it to withstand errors. Here we consider the most robust small neutral networks: the graphs…

We derive eigenvalue bounds for the $t$-distance chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo,…