Related papers: Minimizing Laplacian spectral radius of unicyclic …

For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $\kappa^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$…

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix…

In this paper, we determine the two normalized Laplacian spectrum of generalized subdivision- vertex corona, subdivision-edge corona for a connected regular graph with an arbitrary reg- ular graph in terms of their normalized Laplacian…

Let $\Gamma=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $gr(G)$ the girth of $G$, which is the length of the shortest cycle in $G$. Let $r(G,\sigma)$ be the rank of $(G,\sigma)$. In this paper, we will…

Let $t\geq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $\mu\left( G\right) $ satisfies \[ \mu\left( G\right) \leq\frac{t-1}{2}+\sqrt{n+\frac{t^{2}-2t-3}{4}}, \] with equality if and…

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of order (square root of n) noncospectral unicyclic…

For $0\leq \alpha < 1$, the $\mathcal{A}_{\alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_{\alpha}(G):=\alpha \mathcal{D}(G)+(1-\alpha) \mathcal{A}(G)$, where…

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and…

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. Obviously, ${\rm…

Let $\lambda^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow…

The minimum skew rank of a simple graph G over the field of real numbers, is the smallest possible rank among all real skew-symmetric matrices whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero…

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the…

Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $\kappa(G) \leq k$ and showed that among those graphs, the…

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least…

Let $d_{v_{i}}$ be the degree of the vertex $v_{i}$ of $G$. The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle…

Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of…

Given a connected graph $G$ on $n$ vertices, with minimum degree $\delta\geq 2$ and girth at least $g \geq 4$, what is the maximum radius $r$ this graph can have? Erd\H{o}s, Pach, Pollack and Tuza established in the triangle-free case…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

Let $G_{n,\gamma}$ be the set of all connected graphs on $n$ vertices with domination number $\gamma$. A graph is called a minimizer graph if it attains the minimum spectral radius among $G_{n,\gamma}$. Very recently, Liu, Li and Xie…