Related papers: Laplacian eigenvalues of independence complexes vi…

As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let $X$ be a simplicial complex on vertex set $V$ of size $n$, and let $X(k)$ denote the…

For a graph $G$ and a non-zero real number $\alpha$, the graph invariant $S_{\alpha}(G)$ is the sum of the $\alpha^{th}$ power of the non-zero signless Laplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of…

We study the eigenspace of the Laplacian matrix of a simple graph corresponding to the largest eigenvalue, subsequently arriving at the theory of modular decomposition of T. Gallai.

In [Graphs Combin.~24 (2008) 469--483.], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+\kappa(G)+(k-2)(\alpha(G)-1)$, then $G$ contains a Hamiltonian cycle,…

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the…

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2\Delta(G)$, where $\Delta(G)$ is the largest degree of $G$, if and only if $G$ is…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main…

For a connected graph $\mathcal{G}=(V,E)$ with $n$ nodes, $m$ edges, and Laplacian matrix $\boldsymbol{{\mathit{L}}}$, a grounded Laplacian matrix $\boldsymbol{{\mathit{L}}}(S)$ of $\mathcal{G}$ is a $(n-k) \times (n-k)$ principal submatrix…

If K,G are finite abstract simplicial complexes and K is a subcomplex of G then the eigenvalues of the Hodge Laplacian of K are smaller or equal than the eigenvalues of the Hodge Laplacian of G, provided the eigenvalue lists are padded…

Let $X$ be a pure $d$-dimensional simplicial complex. For $0\le k\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let…

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is…

The independence polynomial $I(G,x)$ of a finite graph $G$ is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists…

In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and…

Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\zeta(v)$ as $\zeta(v)={\max}_{H: v\in H}~\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show…

Let $G$ be a finite group of order $n$ and for $1 \leq i \leq k+1$ let $V_i=\{i\} \times G$. Viewing each $V_i$ as a $0$-dimensional complex, let $Y_{G,k}$ denote the simplicial join $V_1*\cdots*V_{k+1}$. For $A \subset G$ let $Y_{A,k}$ be…

For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the…

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho^{L}_{1}\geq \rho^{L}_{2}\geq \cdots \geq \rho^{L}_{n}$, the distance Laplacian energy $DLE(G)$ is defined as…

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We…