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Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be…

Let $G$ be a simple graph or hypergraph, and let $A(G),L(G),Q(G)$ be the adjacency, Laplacian and signless Laplacian tensors of $G$ respectively. The largest $H$-eigenvalues (resp., the spectral radii) of $L(G),Q(G)$ are denoted…

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

The (torsion) complexity of a finite signed graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When $G$ is $d$-periodic (i.e., $G$ has a free ${\mathbb Z}^d$-action by graph…

Let $G$ be a connected graph of order $n$ with $n\geq25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the…

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not…

The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and adding a new edge between two new vertices which lie on adjacent edges of $G$. In this…

Let $\alpha(G)$ and $\beta(G)$, denote the size of a largest independent set and the clique cover number of an undirected graph $G$. Let $H$ be an interval graph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $\phi(G,H)$ denote the…

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We denote by $tF$ the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku and Vizer in 2019 determined the…

Brouwer's Conjecture states that, for any graph $G$, the sum of the $k$ largest (combinatorial) Laplacian eigenvalues of $G$ is at most $|E(G)| + \binom{k+1}{2}$, $1 \leq k \leq n$. We present several interrelated results establishing…

The number of proper $q$-colorings of a graph $G$, denoted by $P_G(q)$, is an important graph parameter that plays fundamental role in graph theory, computational complexity theory and other related fields. We study an old problem of Linial…

We report our experiments in identifying large bipartite subgraphs of simple connected graphs which are based on the sign pattern of eigenvectors belonging to the extremal eigenvalues of different graph matrices: adjacency, signless…

Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its…

Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $\alpha\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_\alpha(G)$ as…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…

We say a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = |V(\mathcal{G})|$ and $\{e \cap S: e \in E(\mathcal{H})\}$ contains $\mathcal{G}$…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real…

Zykov showed in 1949 that among graphs on $n$ vertices with clique number $\omega(G) \le \omega$, the Tur\'an graph $T_{\omega}(n)$ maximizes not only the number of edges but also the number of copies of $K_t$ for each size $t$. The problem…

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $\alpha_r(G)$ is the largest size of a…

For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_\alpha(G) \) matrix, proposed by Nikiforov, is…