Narrow Results

Let $G=(V_G,E_G)$ be a graph with an adjacency matrix $A_G$ and a diagonal degree matrix $D_G$. For any graph $G$ and a real number $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of $G$, denoted by $A_{\alpha }(G)$, is defined as $A_{\alpha }(G)=\alpha D_G+(1-\alpha )A_G$. The generalized subdivision graph $S_G(n_1,m_1)$, derived from the subdivision graph of $G$ having the vertex set $V_G\cup E_G$, comprises a vertex set $V_G\times \{1,2,\dots ,n_1\}\cup E_G\times \{1,2,\dots ,m_1\}$. This construction includes $n_1$ replicas of $V_G$ and $m_1$ replicas of $E_G$, with edges established between vertices $(v,i)$ and $(e,j)$ where $e\in E_G$ is incident to $v\in V_G$ in $G$. In this paper, we derive the $A_{\alpha }$-characteristic polynomial of $S_G(n_1,m_1)$. We demonstrate that if $G$ is a regular graph, then the $A_{\alpha }$-spectrum of $S_G(n_1,m_1)$ is completely determined by the Laplacian spectrum of $G$. Specifically, when $n_1=m_1$, the $A_{\alpha }$-spectrum of $S_G(n_1,m_1)$ is completely determined by the Laplacian spectrum of the subdivision graph of $G$. In conclusion, as an application, we present the construction of infinite families of non-isomorphic graphs that are $A_{\alpha }$-cospectral.

Let $G$ be a simple, connected undirected graph with $m$ vertices and $n$ edges. Let vertex coloring $c$ of a graph $G$ be a mapping $c:V(G)\to S$, where $\left|S\right|=k$ and it is $k$-colorable. Vertex coloring is proper if none of the any two neighboring vertices receives the similar color. An $r$-dynamic coloring is a proper coloring such that $|c(Nbd(v))|\ge$ min$\{r,{\mathrm{\text{deg}}}_G(v)\}$, for each $v\in$$V(G)$. The $r$-dynamic chromatic number of a graph $G$ is the minutest coloring $k$ of $G$ which is $r$-dynamic k-colorable and denoted by ${\chi }_r(G)$. By a simple view, we exhibit that ${\chi }_r(G)\le {\chi }_{r+1}(G)$, howbeit ${\chi }_{r+1}(G)-{\chi }_r(G)$ cannot be arbitrarily small. Thus, finding the result of ${\chi }_r(G)$ is useful. This study gave the result of $r$-dynamic chromatic number for the central graph, Line graph, Subdivision graph, Line of subdivision graph, Splitting graph and Mycielski graph of the Flower graph $F_n$ denoted by $C(F_n)$, $L(F_n)$, $S(F_n)$, $L(S(F_n))$, $S(F_n)$ and $\mu (F_n),$ respectively.

Topological indices are numeric quantities that transform chemical structure to real number. Topological indices are used in QSAR/QSPR studies to correlate the bioactivity and physiochemical properties of molecule. In this paper, some newly designed neighborhood degree-based topological indices named as neighborhood Zagreb index ($M_N$), neighborhood version of Forgotten topological index ($F_N$), modified neighborhood version of Forgotten topological index ($F_N^{\ast }$), neighborhood version of second Zagreb index ($M_2^{\ast }$) and neighborhood version of hyper Zagreb index ($H{M}_N$) are obtained for Graphene and line graph of Graphene using subdivision idea. In addition, these indices are compared graphically with respect to their response for Graphene and line graph of subdivision of Graphene.

The degree-based topological indices are numerical graph invariants that are used to link a molecule’s structural characteristics to its physical, and chemical characteristics. In the investigation, and study of the structural features of a chemical network, it has emerged as one of the most potent mathematical techniques. In this paper, we study the degree-based topological invariants, called K-Banhatti indices, of the line graphs of some rooted product graphs namely, $C_n\{P_{k+1}\}$, $C_n\{S_{m+1}\}$, and ith vertex rooted product graph $C_{i,r}\{P_{k+1}\}$ which are derived by the concept of subdivision.