Let L n be a linear hexagonal chain with n hexagons. Let Lˆ2 n be the graph obtained by the strong prism of a linear hexagonal chain with n hexagons, i.e. the strong product of L n and K 2. In this paper, explicit expressions for degree-Kirchhoff index and number of spanning trees of Lˆ2 n are determined, respectively. Furthermore, it is interesting to find that the degree-Kirchhoff index of Lˆ2 n is almost one eighth of its Gutman index.
The $(k+1)$-core of a graph $G$, denoted by $C_{k+1}(G)$, is the subgraph obtained by repeatedly removing any vertex of degree less than or equal to $k$. $C_{k+1}(G)$ is the unique induced subgraph of minimum degree larger than $k$ with a maximum number of vertices. For $1\leq k\leq m\leq n$, we denote $R_{n, k, m}=K_k\vee(K_{m-k}\cup {I_{n-m}})$. In this paper, we prove that $R_{n, k, m}$ obtains the maximum spectral radius and signless Laplacian spectral radius among all $n$-vertex graphs whose $(k+1)$-core has at most $m$ vertices. Our result extends a recent theorem proved by Nikiforov [Electron. J. Linear Algebra, 27:250--257, 2014]. Moreover, we also present the bipartite version of our result.
The eccentricity matrix ε(G) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The ε-eigenvalues of a graph G are those of its eccentricity matrix ε(G). Wang et al [22] proposed the problem of determining the maximum ε-spectral radius of trees with given order. In this paper, we consider the above problem of n-vertex trees with given diameter. The maximum ε-spectral radius of n-vertex trees with fixed odd diameter is obtained, and the corresponding extremal trees are also determined. The trees with least ε-eigenvalues in [−2 √ 2, 0) have been known. Finally, we determine the trees with least ε-eigenvalues in [−2 − √ 13, −2 √ 2).
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