For positive numbers j and k, an L( j, k)-labeling f of G is an assignment of numbers to vertices of G such thatThe span of f is the difference between the maximum and the minimum numbers assigned by f . The L( j, k)-labeling number of G, denoted by λ j,k (G), is the minimum span over all L( j, k)-labelings of G. In this article, we completely determine the L( j, k)-labeling number (2 j ≤ k) of the Cartesian product of path and cycle.

Let G be a k-connected irregular graph of order n, size m and maximum degree ∆. Let q 1 be the signless Laplacian spectral radius of G. In this article, we prove the following lower bound on 2∆ − q 1 : 2∆ − q 1 > 2(n∆ − 2m)k 2 2(n∆ − 2m)(n 2 − (∆ − k + 2)(n − k)) + nk 2. Moreover, we determine similar bounds for the signless Laplacian spectral radius of proper spanning subgraphs and k-edge-connected graphs.

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An edge set S of a connected graph G is called an anti-Kekulé set if G−S is connected and has no perfect matchings, where G − S denotes the subgraph obtained by deleting all edges in S from G. The anti-Kekulé number of a graph G, denoted by ak(G), is the cardinality of a smallest anti-Kekulé set of G. It is NP-complete to find the smallest anti-Kekulé set of a graph. In this paper, we show that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekulé number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to find all smallest anti-Kekulé sets of a connected cubic graph.

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