The spread of a graph is defined as the difference between the maximum and minimum eigenvalues of its adjacency matrix. A graph has the Hereditarily Bounded Property \(P_{t,r}\) if, for any subgraph H of order at least t, the size of H is at most \(t|V(H)|+r\) , where |V(H)| denotes the number of vertices in H. In this paper, we determine the unique graph with maximum spread among all graphs of order sufficiently large that satisfy the Hereditarily Bounded Property \(P_{t,r}\) , where t is a positive integer and r is a real number with \(r\ge -{t+1\atopwithdelims ()2}\) . Key applications of this result include the characterization of extremal graphs with maximum spread for the following families of asymptotically large graphs: graphs whose size equals their order plus a fixed non-negative integer; graphs excluding a \((t + 1)\) -matching; graphs containing no t disjoint cycles. We conclude by proposing two open problems related to extremal spread properties, inviting further exploration in spectral graph theory.
