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Let φ(L(G))=det (xI−L(G))=∑k=0n(−1)kck(G)xn−k$\phi (L(G)) = \det (xI - L(G)) = \sum\nolimits_{k = 0}^n {( - 1)^k c_k (G)x^{n - k} } $ be the Laplacian characteristic polynomial of G. In this paper, we characterize the minimal graphs with the minimum Laplacian coefficients in 𝒢n,n+2(i) (the set of all tricyclic graphs with fixed order n and matching number i). Furthermore, the graphs with the minimal Laplacian-like energy, which is the sum of square roots of all roots on ϕ(L(G)), is also determined in 𝒢n,n+2(i).

For a k-graph H = (V(H), E(H)), let B(H) be its incidence matrix, and Q(H) = B(H)B(H)T be its signless Laplacian matrix, and this name comes from the fact that Q(H) is exactly the well-known signless Laplacian matrix for 2-graph. Define the largest eigenvalue ρ(H) of Q(H) as the spectral radius of H. In this paper, we give some lower and upper bounds on ρ(H) using some structural parameters (such as independent number, maximum degree, minimum degree, diameter, and so on) of H, which extended or improved some known results. [ABSTRACT FROM AUTHOR]

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) \ Cn.

Let H = (V, E) be a hypergraph and B its incidence matrix. Let Q(H) = BBT be the signless Laplacian matrix of H and λ1(Q), λ2(Q), ..., λn(Q) are its eigenvalues. The signless Laplacian Estrada index of H is defined as SLEEH=∑i=1neλiQ which is first extended to hypergraph. We obtain lower and upper bounds for the index in terms of the number of vertices and edges of H. We also determine the unique graph with maximum SLEE among all k‐uniform hypergraphs. In addition, we characterize the extremal hypertrees with the smallest and the largest SLEE among all k‐uniform hypertrees. [ABSTRACT FROM AUTHOR]

In wireless networks, it is still a challenge to allocate the limited resources among users to meet their specific quality of service (QoS) requirements, especially when the users have different traffic types, i.e., the hard QoS traffic, the best effort traffic, and the soft QoS traffic. In this paper, we develop the utility-based resource allocation algorithms in the following three tasks: 1) resource allocation among the hard QoS traffic and the soft QoS traffic; 2) resource allocation among the best effort traffic and the soft QoS traffic; and 3) finally resource allocation among the hard QoS traffic, the best effort traffic, and the soft QoS traffic, by solving the network utility maximization problem using the Karush–Kuhn–Tucker condition. We develop a number of critical theorems to give the conditions that find the optimal solutions for the above three cases in a unified framework. These theorems then act as design guidelines for the three algorithms. The proposed algorithms take into account the traffic type, the total available resources and the users’ channel qualities. We evaluate the time complexity of the proposed algorithms, which comes out to be polynomial, and study the network performance by numerical examples. Numerical results demonstrate the bandwidth allocations, the fairness index and the total maximum utility under different channel qualities and resource situations. [ABSTRACT FROM PUBLISHER]

Optimization theory and nonlinear programming method have successfully been applied into wire-lined networks (e.g., the Internet) in developing efficient resource allocation and congestion control schemes. The resource (e.g., bandwidth) allocation in a communication network has been modeled into an optimization problem: the objective is to maximize the source aggregate utility subject to the network resource constraint. However, for wireless networks, how to allocate the resource among the soft quality of service (QoS) traffic remains an important design challenge. Mathematically, the most difficult comes from the non-concave utility function of soft QoS traffic in the network utility maximization (NUM) problem. Previous result on this problem has only been able to find its sub-optimal solution. Facing this challenge, this paper establishes some key theorems to find the optimal solution and then present a complete algorithm called utility-based allocation for soft QoS to obtain the desired optimal solution. The proposed theorems and algorithm act as designing guidelines for resource allocation of soft QoS traffic in a wireless network, which take into account the total available resource of network, the users' traffic characteristics, and the users' channel qualities. By numerical examples, we illustrate the explicit solution procedures. [ABSTRACT FROM AUTHOR]

ABSTRACT The resource (bandwidth) allocation in a network is usually casted into a so-called network utility maximization (NUM) problem, which solution strategy has successfully generated distributed algorithms for congestion controlling of elastic traffic in a wire-lined network. However, for resource allocation of inelastic traffic including soft QoS (quality of service) traffic in a wireless network, this approach still faces challenges. First, it is hard for the wireless system to dynamically model the utility function of the users. Second, the utility function of soft QoS traffic is usually nonconcave, which brings the NUM optimization problem to be mathematically intractable. With deviation to the usual NUM theory, this paper proposes a novel optimization model and its algorithm to allocate bandwidth around the user's desired value to the soft QoS traffic in a wireless network. Our approach takes advantage of the basic feature of soft QoS traffic; that is, it demands a preferred amount of bandwidth but allows some flexibility during normal operation. Compared with the utility-based approaches and solutions, our approach avoids the difficulty of finding the exact utility function expression for each user by using the preference bandwidth value. This facilitates the operation of real wireless networks. The proposed model and algorithm are verified by an example, which demonstrate better performance than the NUM approach.Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Wiener index of a connected k‐uniform hypergraph is defined as the summation of distances between all pairs of vertices. We determine the unique k‐uniform unicyclic hypergraphs with maximum and second maximum, minimum and second minimum Wiener indices, respectively. [ABSTRACT FROM AUTHOR]