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A complete characterization of spectra of the Randić matrix of level-wise regular trees | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 14، دوره 11، شماره 2، شهریور 2026، صفحه 539-562 اصل مقاله (502.29 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29304.1936 | ||
| نویسندگان | ||
| Punit Vadher1؛ Bantva Devsi* 2 | ||
| 1Gujarat Technogical University, Ahmedabad - 382 424, Gujarat, India | ||
| 2Lukhdhirji Engineering College, Morvi - 363 642, Gujarat, India | ||
| چکیده | ||
| Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$ and $d_i$ be the degree of the vertex $v_i$. The Randić matrix ${\bf R}(G) = [r_{i,j}]$ of graph $G$ is an $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} = 1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A level-wise regular tree is a tree rooted at one vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0 \leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randić matrix of level-wise regular trees. We prove that the eigenvalues of the Randić matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence $(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees. | ||
| کلیدواژهها | ||
| graph spectrum؛ Randić matrix؛ level-wise regular tree | ||
| مراجع | ||
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