Related papers: A graph energy conjecture through the lenses of se…

A subgraph $H$ of a multigraph $G$ is overfull if $ |E(H) | > \Delta(G) \lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph version of the…

A $\mathbb{T}$-gain graph, $\Phi = (G, \varphi)$, is a graph in which the function $\varphi$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency…

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…

Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $\Delta$. Let $\lambda_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the…

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we present a new method to compare the energies of two $k$-subdivision bipartite graphs on some cut edges. As the…

Let $G$ be a simple graph of order $n$. The energy $E(G)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $G$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose…

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n…

For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denoted by $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. We call a family…

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph…

For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $\mu(G)$ is the number of edges in a maximum matching of $G$, while $\Delta$ is the maximum vertex…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$…

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be…

Let $G$ be a graph on $n$ vertices with $r := \lfloor n/2 \rfloor$ and let $\lambda_1 \geq...\geq \lambda_{n} $ be adjacency eigenvalues of $G$. Then the H\"uckel energy of $G$, HE($G$), is defined as $$\he(G) = {ll} 2\sum_{i=1}^{r}…

The resolvent energy of a graph $G$ of order $n$ is defined as $ER=\sum_{i=1}^n (n-\lambda_i)^{-1}$, where $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $G$. In a recent work [Gutman et al., {\it MATCH Commun. Math. Comput.…

For a simple graph $G=(V,E)$, a \emph{proper total weighting} is a mapping $w: V\cup E\rightarrow \mathbb R$ such that for every edge $uv\in E$, $w(u)+\sum_{e\ni u}w(e)\neq w(v)+\sum_{e\ni v}w(e)$. The graph $G$ is said…

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the…

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for…

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum…

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq…

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$.…