Graphical invariants play a fundamental role in characterizing the structural and distance-related properties of complex networks, with significant applications in graph theory, chemical graph theory, and communication systems. Among these, the distance eccentric connectivity index (DECI) and the $ d $-connectivity index (DCI) have emerged as effective descriptors that integrate vertex degree, eccentricity, and distance information. In this paper, we present an exact analytical study of these two distance-based graphical invariants for several families of circulant networks. Closed-form expressions are established for the DECI and DCI of 3-, 4-, and 5-regular circulant networks. Furthermore, unified general formulas for the DCI are derived for generalized circulant networks with arbitrary even and odd regularities. The obtained results reveal the influence of graph order, regularity, and parity on the behavior of these invariants, providing a comprehensive characterization of their structural properties. The proposed formulas not only extend the theoretical understanding of distance-based graphical invariants for highly symmetric networks but also offer efficient computational tools for the analysis of communication networks, chemical graph structures, and other complex, interconnected systems.
Citation: Haoyu Kong, Muhammad Salman, Tahira Noreen, Hanen Karamti, Ziye Kong. Distance eccentric connectivity and $ d $-connectivity indices of circulant networks: exact formulas and generalizations[J]. AIMS Mathematics, 2026, 11(11): 36100-36118. doi: 10.3934/math.20261398
Graphical invariants play a fundamental role in characterizing the structural and distance-related properties of complex networks, with significant applications in graph theory, chemical graph theory, and communication systems. Among these, the distance eccentric connectivity index (DECI) and the $ d $-connectivity index (DCI) have emerged as effective descriptors that integrate vertex degree, eccentricity, and distance information. In this paper, we present an exact analytical study of these two distance-based graphical invariants for several families of circulant networks. Closed-form expressions are established for the DECI and DCI of 3-, 4-, and 5-regular circulant networks. Furthermore, unified general formulas for the DCI are derived for generalized circulant networks with arbitrary even and odd regularities. The obtained results reveal the influence of graph order, regularity, and parity on the behavior of these invariants, providing a comprehensive characterization of their structural properties. The proposed formulas not only extend the theoretical understanding of distance-based graphical invariants for highly symmetric networks but also offer efficient computational tools for the analysis of communication networks, chemical graph structures, and other complex, interconnected systems.
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Figures(5) / Tables(6)
Haoyu Kong, Muhammad Salman, Tahira Noreen, Hanen Karamti, Ziye Kong. Distance eccentric connectivity and $ d $-connectivity indices of circulant networks: exact formulas and generalizations[J]. AIMS Mathematics, 2026, 11(11): 36100-36118. doi: 10.3934/math.20261398
Preliminaries and basic results
The circulant network $ C_{12}(1, \tfrac{12}{2}) $ with the distance $ d(v_1, v) $ written inside each vertex. The fixed vertex $ v_1 $ is black, the eccentric vertices (at distance $ \mathrm{ecc} = 3 $) are red, and the $ d $-neighbors (at distance $ d = 3 $) have a thick blue border. Here, $ \deg^{De}(v) = 4 $, so $ \xi^{De}(G) = 12\cdot 4\cdot 3 = 144 $ (Theorem 2), and $ n_d(v) = 4 $, so $ C_d(G) = 12\cdot 4\cdot 3 = 144 $ (Theorem 3)
The circulant network $ C_{17}(1, 2) $ with the distance $ d(v_1, v) $ written inside each vertex. The fixed vertex $ v_1 $ is black, the eccentric vertices (at distance $ \mathrm{ecc} = 4 $) are red, and the $ d $-neighbors (at distance $ d = 4 $) have a thick blue border. Here, $ \deg^{De}(v) = 4 $, so $ \xi^{De}(G) = 17\cdot 4\cdot 4 = 272 $ (Theorem 5), and $ n_d(v) = 4 $, so $ C_d(G) = 17\cdot 4\cdot 4 = 272 $ (Theorem 6)
The circulant network $ C_{16}(1, 2, \tfrac{16}{2}) $ with the distance $ d(v_1, v) $ written inside each vertex. The fixed vertex $ v_1 $ is black, and the eccentric vertices (at distance $ \mathrm{ecc} = 3 $) are red. Here, $ \deg^{De}(v) = 2 $, so $ \xi^{De}(G) = 16\cdot 2\cdot 3 = 96 $ (Theorem 7). Because $ d(v) = 5 > \mathrm{ecc}(v) $, there is no $ d $-neighbor, $ n_d(v) = 0, $ and $ C_d(G) = 0 $ (Theorem 8)
Behavior of the two indices on the three circulant networks, computed from Theorems 2–8. (a) $ \xi^{De}(G)/n^{2} $: values group into levels determined by the residue class of $ n $ (mod 4 or mod 8). (b) $ C_d(G)/n $: index vanishes below a family-dependent threshold and then stabilizes at $ 12 $, $ 16, $ and $ 40 $ for $ C_n(1, n/2) $, $ C_n(1, 2), $ and $ C_n(1, 2, n/2), $ respectively
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