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In light of the successful investigation of the adjacency matrix, a significant amount of its modification is observed employing numerous topological indices. The matrix corresponding to the well-known first Zagreb index is one of them. The entries of the first Zagreb matrix are d u i + d u j , if u i is connected to u j ; 0, otherwise, where d u i is degree of i-th vertex. The current work is concerned with the mathematical properties and chemical significance of the spectral radius ( ρ 1 ) associated with this matrix. The lower and upper bounds of ρ 1 are computed with characterizing extremal graphs for the class of unicyclic graphs and trees. The chemical connection of the first Zagreb spectral radius is established by exploring its role as a structural descriptor of molecules. The isomer discrimination ability of ρ 1 is also explained. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix $ A(G) $ A (G). Let $ \tau (G) $ τ (G) denote the smallest positive eigenvalue of $ A(G) $ A (G). In 1990, Pavlíková and Kr $ \breve{c} $ c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let $ \mathscr {U} $ U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each $ m\geq ~3 $ m ≥ 3 , let $ \mathscr {U}_n $ U n be the subclass of $ \mathscr {U} $ U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in $ \mathscr {U} $ U such that $ \tau (U)\leq \sqrt {2}-1 $ τ (U) ≤ 2 − 1. We then find the unique graph $ U_o^n $ U o n (resp. $ U_e^n $ U e n ) having the maximum τ value among all graphs in $ \mathscr {U}_n $ U n when m is odd (resp. when m is even). Finally, we prove that $ U_o^6 $ U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in $ \mathscr {U} $ U . As a consequence, we obtain a sharp upper bound for $ \tau (U) $ τ (U) when $ U\in \mathscr {U} $ U ∈ U . [ABSTRACT FROM AUTHOR]