Abstract. Given two graphs G 1 , with vertices 1, 2, ..., n and edges e 1 , e 2 , ..., em, and G 2 , the edge corona G 1 ⋄ G 2 of G 1 and G 2 is defined as the graph obtained by taking m copies of G 2 and for each edge e k = ij of G, joining edges between the two end-vertices i, j of e k and each vertex of the k-copy of G 2 . In this paper, the adjacency spectrum and Laplacian spectrum of G 1 ⋄ G 2 are given in terms of the spectrum and Laplacian spectrum of G 1 and G 2 , respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

No abstract

In this paper, we give sufficient conditions on the spectral radius for a bipartite graph being Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also provide tight sufficient conditions on the signless Laplacian spectral radius for a graph to be Hamiltonian and traceable, which improve the results of Yu and Fan (2012) [11].

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f : E → {1, . . . , |E|} such that for any pair of adjacent vertices x and y, ff (e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions for χ la (H) ≤ χ la (G) are obtained, where H is obtained from G with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.

A total labeling of a graph G = (V, E) is said to be local total antimagic if it is a bijection f : V ∪ E → {1, . . . , |V | + |E|} such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex v, w f (v) =f (e) with e ranging over all the edges incident to v, and the induced weight of an edge uv is w f (uv) = f (u) + f (v). The local total antimagic chromatic number of G, denoted by χ lt (G), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of G. In this paper, we first obtained general lower and upper bounds for χ lt (G) and sufficient conditions to construct a graph H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1}. We then completely characterized graphs G with χ lt (G) = 3. Many families of (disconnected) graphs H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1} are also obtained.