A k-tree is a tree with maximum degree at most k. In this paper, we give a degree sum condition for a graph to have a spanning k-tree in which specified vertices have degree less than k. We denote by σk(G) the minimum value of the degree sum of k independent vertices in a graph G. Let k ≥ 3 and s ≥ 0 be integers, and suppose G is a connected graph and σk(G) ≥ |V (G)|+s−1. Then for any s specifi...

In this work, we obtain the following new results: – Given a tree T = (V, E) with a length function ` : E → R and a weight function w : E → R, a positive integer k, and an interval [L, U ], the Weight-Constrained k Longest Paths problem is to find the k longest paths among all paths in T with weights in the interval [L,U ]. We show that the Weight-Constrained k Longest Paths problem has a lower...

Let rk(n) denote the number of representations of n as a sum of k squares and tk(n) the number of representations of n as a sum of k triangular numbers. We give an elementary, combinatorial proof of the relations rk(8n+ k) = cktk(n), 1 ≤ k ≤ 7, where c1 = 2, c2 = 4, c3 = 8, c4 = 24, c5 = 112, c6 = 544 and c7 = 2368.

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

let $m_r$ be a module with $s=end(m_r)$. we call a submodule $k$ of $m_r$ annihilator-small if $k+t=m$, $t$ a submodule of $m_r$, implies that $ell_s(t)=0$, where $ell_s$ indicates the left annihilator of $t$ over $s$. the sum $a_r(m)$ of all such submodules of $m_r$ contains the jacobson radical $rad(m)$ and the left singular submodule $z_s(m)$. if $m_r$ is cyclic, then $a_r(m)$ is the unique ...

A simple q-analogue of the sum of cubes is given. This answers a question posed in this journal by Garrett and Hummel. The sum of cubes and its q-analogues It is well-known that the first n consecutive cubes can be summed in closed form as n ∑ k=1 k = ( n + 1 2 )2 . Recently, Garrett and Hummel discovered the following q-analogue of this result: n ∑ k=1 qk−1 (1− q)(2− qk−1 − q) (1− q)2(1− q2) =...

Some new properties of the eigenvalues of the subdirect sums are presented for the particular case of 1-subdirect sums. In particular, it is shown that if an eigenvalue λ is associated with certain blocks of matrix A or matrix B then λ is also an eigenvalue associated with the 1-subdirect sum A ⊕1 B. Some results concerning eigenvectors of the k-subdirect sum A⊕k B for an arbitrary positive int...

A set S of integers is said to be sum-free if a, b e 5 implies a + b 6 S. In this paper, we investigate two new problems on sum-free sets: (1) Let f(k) denote the largest positive integer for which there exists a partition of (1, 2,... ,f(k)) into k sum-free sets, and let h(k) denote the largest positive integer for which there exists a partition of {1, 2, . . . , h(k)) into k sets which are su...

Conditions are given which guarantee that the k-subdirect sum of S-strictly diagonally dominant matrices (S-SDD) is also S-SDD. The same situation is analyzed for SDD matrices. The converse is also studied: given an SDD matrix C with the structure of a k-subdirect sum and positive diagonal entries, it is shown that there are two SDD matrices whose subdirect sum is C. AMS subject classifications...