Narrow Results

The non-commuting graph of a non-abelian group $G$ with center $Z(G)$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x,y$ are adjacent if $xy\ne yx$. In this paper, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of certain finite non-abelian groups. We obtain several conditions such that the non-commuting graph of $G$ is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the hyperenergetic and hypoenergetic properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total $\pi$-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

For a graph $G$ with $n$ vertices and $m$ edges, the energy (or the adjacency energy) is the sum of the absolute values of the eigenvalues (or adjacency eigenvalues) of $G$, that is, ${ℰ}(G)={\sum }_{i=1}^n|{\lambda }_i(G)|$, where ${\lambda }_i(G)$ is the $i\mathrm{\text{th}}$ adjacency eigenvalue of $G$. In this paper, we discuss the characteristic polynomial and the energy of dendrimers. We first show the formula obtained for the characteristic polynomial and the energy of tree dendrimers considered in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045] are incorrect. Using a different algorithmic procedure we will derive the correct formula for the characteristic polynomial and the energy of dendrimer $d(3,k)$. Further, we obtain the energy of $d(l,k)$ for $k=4$ and $k=5$, which partially answers a problem raised in Bokhary and Tabassum [The energy of some tree dendrimers, J. Appl. Math. Comput.68 (2022) 1033–1045].

The distance signless Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of the distance signless Laplacian matrix of $G$, defined as $D^Q(G)=\mathrm{\text{Tr}}(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $\mathrm{\text{Tr}}(G)$ is the diagonal matrix of vertex transmissions of $G$. In this paper, we determine some bounds on the distance signless Laplacian spectral radius of $G$ based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math.37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.

For a graph $G$ with vertex set $V_G=\{v_1,v_2,\dots ,v_n\}$ and edge set $E_G=\{e_1,e_2,\dots ,e_m\}$, let $S_G$ denotes the subdivision graph of $G$ with vertex set $V_G\cup E_G$. In $S_G$, replace each vertex $v_i$, $i=1,2,\dots ,n$, by $n_1$ vertices and join every vertex to the neighbors of $v_i$. Then in the resulting graph, replace each vertex $e_j$, $j=1,2,\dots ,m$, by $m_1$ vertices and join every vertex to the neighbors of $e_j$. The resulting graph is denoted by $S_G(n_1,m_1)$. This generalizes the construction of the subdivision graph $S_G$ to $S_G(n_1,m_1)$ of a graph $G$. In this paper, we provide the complete information about the spectrum of $S_G(n_1,m_1)$ using the spectrum of $S_G$. Further, we determine the Laplacian spectrum of $S_G(n_1,m_1)$ using the Laplacian spectrum of $G$, when $G$ is a regular graph. Also, we find the Laplacian spectrum of $S_G(n_1,m_1)$ using the Laplacian spectrum of $S_G$ when $n_1=m_1$. The energy of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. The incidence energy of a graph $G$ is defined as the sum of the square roots of the signless Laplacian eigenvalues of $G$. Finally, as an application, we show that the energy of the graph $S_G(n_1,m_1)$ is completely determined by the incidence energy of the graph $G$.

Let $n\ge 1$ and $U_n$ be the multiplicative group of the additive group ${\mathbb{Z}}_n$ of integers modulo $n$. The unitary Haar graph $H_n=\mathrm{\text{Haar}}({\mathbb{Z}}_n,U_n)$ is a graph with vertex set ${\mathbb{Z}}_n\times \{0,1\}$ and the edge set $\{\{(a,0),(b,1)\}|b-a\in U_n\}$. In this paper, we study some structural properties of $H_n$ such as total chromatic number, diameter, planarity, the girth, vertex-connectivity and edge-connectivity. Also, some spectral properties of the graph $H_n$ are determined.

Let $\mathrm{\Gamma }$ be a simple graph of order $n$ having $m$ edges and let $\overrightarrow{\mathrm{\Gamma }}$ be an orientation of $\mathrm{\Gamma }$. The hermitian skew Laplacian matrix $SL(\overrightarrow{\mathrm{\Gamma }})$ of $\overrightarrow{\mathrm{\Gamma }}$ is defined as $SL(\overrightarrow{\mathrm{\Gamma }})=\overline{D}(\overrightarrow{\mathrm{\Gamma }})-iS(\overrightarrow{\mathrm{\Gamma }})$, where $i=\sqrt{-1}$ is the imaginary unit, $\overline{D}(\overrightarrow{\mathrm{\Gamma }})$ is the diagonal matrix of oriented degrees ${\overline{d}}_i=d_i^{+}-d_i^{-}$ and $S(\overrightarrow{\mathrm{\Gamma }})$ is the skew matrix. The spectral radius of the matrix $SL(\overrightarrow{\mathrm{\Gamma }})$ is the hermitian skew Laplacian spectral radius and the sum of the absolute values of eigenvalues of $SL(\overrightarrow{\mathrm{\Gamma }})$ is the hermitian skew Laplacian energy of $\overrightarrow{\mathrm{\Gamma }}$. In this paper we study the hermitian spectral radius and hermitian skew Laplacian energy of $\overrightarrow{\mathrm{\Gamma }}$. We obtain some new bounds for the hermitian skew Laplacian spectral radius and characterize the extremal oriented graphs attaining these bounds. Further, we obtain some bounds for the hermitian skew Laplacian energy and locate the extremal oriented graphs.