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We analyze an optimal boundary control problem for heat convection equations in a three-dimensional domain, with mixed boundary conditions. We prove the existence of optimal solutions, by considering boundary controls for the velocity vector and the temperature. The analyzed optimal control problem includes the minimization of a Lebesgue norm between the velocity and some desired field, as well as the temperature and some desired temperature. By using the Lagrange multipliers theorem we derive an optimality system. We also give a second-order sufficient condition.

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In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an (n, m) connected graph G with m > n ≥ 4 the spread of G is less than or equal to the spread of its line graph, where the equality holds if and only if G is regular bipartite. A sufficient condition for the spread of the graph not be greater than the signless Laplacian spread for a class of non bipartite and non regular graphs is proved. Additionally, we derive an upper bound for * Corresponding author the spread of the line graph of graphs on n vertices having a vertex (edge) connectivity less than or equal to a positive integer k. Combining techniques of interlacing of eigenvalues, we derive lower bounds for the Laplacian and signless Laplacian spread of the total graph of a connected graph. Moreover, for a regular graph, an upper and lower bound for the spread of its total graph is given.

A bug Bug p,r1,r2 is a graph obtained from a complete graph K p by deleting an edge uv and attaching the paths P r1 and P r2 by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q 1 (G) be the spectral radius of Q(G). It is known that the bug B 0 = Bug n−d+2,b d 2 c,d d 2 e maximizes q 1 (G) among all graphs G of order n and diameter d. For a bug B of order n and diameter d, n − d is an eigenvalue of Q(B) with multiplicity n − d − 1. In this paper, we prove that remainder d + 1 eigenvalues of Q(B), among them q 1 (B), can be computed as the eigenvalues of a symmetric tridiagonal matrix of order d + 1. Finally, we show that q 1 (B 0 ) can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order d 2 + 1 whenever d is even.