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 revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of ed...

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.

The Randić index R(G) of a graph G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given ...

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

We determine the graph with the largest signless Laplacian spectral radius among all unicyclic graphs with fixed matching number.