Related papers: On graphs with $m(\partial^L_1)=n-3$

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq \cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer such…

The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric…

For a finite simple graph $G$, say $G$ is of dimension $n$, and write $\dim(G) = n$, if $n$ is the smallest integer such that $G$ can be represented as a unit-distance graph in $\mathbb{R}^n$. Define $G$ to be \emph{dimension-critical} if…

In this article, we consider the bipartite graphs $K_2 \times K_n$. We prove that the connectedness of the complex $\displaystyle \text{Hom}(K_2\times K_{n}, K_m) $ is $m-n-1$ if $m \geq n$ and $m-3$ in the other cases. Therefore, we show…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In this…

In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of…

For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are…

It is well known that the complete multipartite graphs can not be determined by their adjacency spectra. But in this paper, we prove that they can be determined by their distance spectra, which confirms the conjecture proposed by Lin, Hong,…

Let $G$ be a simple graph of order $n$. It is known that any Laplacian eigenvalue of $G$ belongs to the interval $[0,n]$. For an interval $I\subseteq [0, n]$, denote by $m_GI$ the number of Laplacian eigenvalues of $G$ in $I$, counted with…

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…

Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\mu_1(G)\ge\cdots\ge\mu_{n-1}(G)\ge\mu_n(G)=0$. It is a conjecture on Laplacian spread of…

A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the…

We show that deciding whether a given graph $G$ of size $m$ has a unique perfect matching as well as finding that matching, if it exists, can be done in time $O(m)$ if $G$ is either a cograph, or a split graph, or an interval graph, or…

The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide…

A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability $p$. One can ask for the probability that all vertices can communicate ({\em all-terminal…

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row…

For a simple graph $G$, the $3$-distance graph, $D_3(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph $G$. For a connected graph $G$, we provide some conditions for…

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the…

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…