### فهرست مطالب

• Volume:3 Issue:3, 2014
• تاریخ انتشار: 1393/03/24
• تعداد عناوین: 6
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• Hong Wenxi, You Lihua Pages 1-9
The sharp upper bounds and the sharp lower bounds of the largest eigenvalues \$lambda_1\$, the least eigenvalue \$lambda_n\$, the second largest eigenvalue \$lambda_2\$, the spread and the separator among all firefly graphs on \$n\$ vertices are determined.
Keywords: Firefly graph, largest eigenvalue, least eigenvalue, spread, separator
• Torsten Sander, Khalida Mohammad Nazzal Pages 11-20
Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T(􀀀(R)),is the simple (undirected) graph with vertex set R, and two distinct vertices are adjacent if their sum is a zero divisor in R. Let T(􀀀(Zn)) be the total graph for the ring of integers modulo n. Minimum zero-sum k-flows for T(􀀀(Zn)) are constructed, in particular, the cases n is even, or n = p^r, n = p^rq^s, where p and q are primes, r and s are positive integers, are considered. Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated.
Keywords: Constant, sum k, flow, minimum flow, the ring of integers modulo n, total graph of a commutative ring, zero, sum k, flow
• Ran Gu, Fei Huang, Xueliang Li Pages 21-33
Let \$G\$ be a simple graph with vertex set \$V(G) = {v_1, v_2,ldots, v_n}\$ and \$d_i\$ the degree of its vertex \$v_i\$, \$i = 1, 2, cdots, n\$. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randic matrix \$textbf{R}_alpha\$ of \$G\$, which is defined by \$(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha\$ if \$v_i\$ and \$v_j\$ are adjacent, and zero otherwise. Similarly, the general Randi'{c} eigenvalues are the eigenvalues of the general Randi'{c} matrix, the greatest general Randi'{c} eigenvalue is the general Randi'{c} spectral radius of \$G\$, and the general Randi'{c} energy is the sum of the absolute values of the general Randi'{c} eigenvalues. In this paper, we prove some properties of the general Randic matrix and obtain lower and upper bounds for general Randi'{c} energy, also, we get some lower bounds for general Randi'{c} spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randi'{c} energy when \$alpha=-1/2\$.[2mm] noindent{bf Keywords:} general Randic matrix, general Randic energy, eigenvalues, spectral radius.
Keywords: general Randic matrix, general Randic energy, eigenvalues, spectral radius