A set S ⊆ V is independent in a graph G = (V,E) if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. A connected graph having only one cycle, say C, is a unicyclic graph. In this paper we prove that if G is a uni...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, th...

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let Un,d be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from Un,d minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hos...

The Wiener index of a graph G is defined as W (G) = ∑ u,v dG(u, v), where dG(u, v) is the distance between u and v in G, and the sum goes over all pairs of vertices. In this paper, we characterize the connected unicyclic graph with minimum Wiener indices among all connected unicyclic graphs of order n and girth g with k pendent vertices.

*Correspondence: [email protected] 2School of Science, Zhejiang A&F University, Hangzhou, 311300, China Full list of author information is available at the end of the article Abstract Let S(G ) be the skew-adjacency matrix of an oriented graph G with n vertices, and let λ1,λ2, . . . ,λn be all eigenvalues of S(G ). The skew-spectral radius ρs(G ) of G is defined as max{|λ1|, |λ2|, . . . , |λn|}....

In the article “Computing the vertex separation of unicyclic graphs”, Information and Computation 192, pp. 123–161, 2004, Ellis et al. proposed an O(n log n) algorithm for computing both the vertex separation and an optimal layout of a unicyclic graph with n vertices. Using the data structures label and label array, we improve the time complexity of their algorithm to O(n).

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let U g n be the set of unicyclic graphs of order n with girth g. For all integers n and g with 5 ≤ g ≤ n − 6, we determine the first ⌊ g2⌋+ 3 spectral radii of unicyclic graphs in the set U g n .

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order n with girth and the matching number β ≥ 3g 2 . Moreover, we characterize all extremal graphs which attain the lower bound.

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.