Related papers: Path-Minimality of $p$-Energy for Connected Graphs

For a given simple graph \( G \), the \( p \)-energy of \( G \), denoted by \( \mathcal{E}_p(G) \), is defined as the sum of the \( p \)-th power of the absolute values of the eigenvalues of its adjacency matrix. Let \( S_n \) denote the…

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and…

For a Hermitian matrix $A$ of order $n$ with eigenvalues $\lambda_1(A)\ge \cdots\ge \lambda_n(A)$, define \[ \mathcal{E}_p^+(A)=\sum_{\lambda_i > 0} \lambda_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{\lambda_i<0} |\lambda_i(A)|^p,\] to be the…

Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1 \geq \cdots \geq\lambda_n$. Let \[s^+(G)=\sum_{\lambda_i>0} \lambda_i^2, \qquad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called…

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $n$ be an even number and $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with maximum…

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai…

Given a graph $G$, we associate a path matrix $P$ whose $(i, j)$ entry represents the maximum number of vertex disjoint paths between the vertices $i$ and $j$, with zeros on the main diagonal. In this note, we resolve four conjectures from…

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been published in…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end…

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $C_n$ denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joining two cycles…

Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $\lambda_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n}…

Let $G$ be a graph with $n$ vertices and $m$ edges. The energy $E$ of the graph $G$ is defined as the sum of the moduli of the adjacency eigenvalues $\lambda_{1} \geq \lambda_{2} \geq \ldots \geq \lambda_{n}$ of $G$: $$…

Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, $\mathcal{E}(G)$ is the sum of all absolute values of the eigenvalues of the adjacency matrix $A(G)$. It is the first eigenvalue-based topological…

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known…

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In…

Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $…

We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

For a connected graph $G$ with order $n$, let $e(G)$ be the number of its distinct eigenvalues and $d$ be the diameter. We denote by $m_G(\mu)$ the eigenvalue multiplicity of $\mu$ in $G$. It is well known that $e(G)\geq d+1$, which shows…