The generalized adjacency-distance matrix of connected graphs.

Let G be a connected graph with adjacency matrix $ A(G) $ A (G) and distance matrix $ \mathcal {D}(G) $ D (G). The adjacency-distance matrix of G is defined as $ S(G) = \mathcal {D}(G) + A(G) $ S (G) = D (G) + A (G). In this paper, $ S(G) $ S (G) is generalized by the convex linear combinations \[ S_{\alpha}(G)=\alpha \mathcal{D}(G)+(1-\alpha)A(G) \] S α (G) = α D (G) + (1 − α) A (G) where $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ]. Let $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) be the spectral radius of $ S_{\alpha }(G) $ S α (G). This paper presents results on $ S_{\alpha }(G) $ S α (G) with emphasis on $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) and some results on $ S(G) $ S (G) are extended to all α in some subintervals of $ [0,1] $ [ 0 , 1 ]. For $ \alpha \in [1/2,1] $ α ∈ [ 1 / 2 , 1 ] , the trees attaining the largest and the smallest $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) among trees of fixed order are determined and it is proved that $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) is a branching index. Moreover, for $ \alpha \in (1/2,1] $ α ∈ (1 / 2 , 1 ] , the graphs that uniquely minimize $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) : among all connected graphs of fixed order and fixed connectivity, and among all connected graphs of fixed order and fixed chromatic number are characterized. [ABSTRACT FROM AUTHOR]