Degree Adjacency Diametrical Matrix of Graphs Spectral Properties and Energy Bounds

Source abstract

Graph matrices that combine structural and metric information have attracted considerable attention in spectral graph theory. In this paper, we introduce the \emph{Degree--Adjacency--Diametrical (DAD) matrix} of a connected graph $G$. The DAD matrix is defined by $$(DAD(G))_{ij}= \begin{cases} d(v_i), & \text{if } i=j, 1, & \text{if } v_i\sim v_j, \operatorname{diam}(G), & \text{if } d(v_i,v_j)=\operatorname{diam}(G), 0, & \text{otherwise}, \end{cases}$$ where $d(v_i)$ denotes the degree of the vertex $v_i$. We establish fundamental spectral properties of the DAD matrix by deriving trace identities, characteristic polynomials, and spectra for several classes of graphs. Furthermore, we obtain a spectral expression for the Wiener index of graphs with diameter two, derive upper bounds for the largest DAD eigenvalue, and establish lower bounds for the DAD energy. Explicit spectral characterizations are also obtained for complete multipartite graphs. The proposed matrix provides a unified spectral framework that simultaneously incorporates degree, adjacency, and diametrical information, thereby enriching the study of graph spectra and introducing a new graph matrix for spectral graph theory.