In this.paper, by joint tree model, we obtain the genera of two types of graphs, which are suspensions of cartesian products of two types of bipartite graphs from a vertex.

Let \(G\) be a connected graph with a perfect matching on \(2n\) vertices (\(n \geq 2\)). A graph \(G’\) is a contraction of \(G\) if it can be obtained from \(G\) by a sequence of edge contractions. Then \(G\) is said to be edge contractible if for any contraction \(G’\) of \(G\) with \(|V(G’)|\) even, \(G’\) has a perfect matching. In this note, we obtain a sufficient and necessary condition for a graph to be an edge contractible graph.

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.

A vertex subset \(S\) of a digraph \(D\) is called a dominating set of \(D\) if every vertex not in \(S\) is adjacent from at least one vertex in \(S\). The domination number of \(D\), denoted by \(\gamma(D)\), is the minimum cardinality of a dominating set of \(D\). We characterize the rooted trees and connected contrafunctional digraphs \(D\) of order \(n\) satisfying \(\gamma(D) = \left\lceil \frac{n}{2}\right\rceil\). Moreover, we show that for every digraph \(D\) of order \(n\) with minimum in-degree at least one, \(\gamma(D) \leq \frac{(k+1)n}{2k+1}\), where \(2k+1\) is the length of a shortest odd directed cycle in \(D\), and we characterize the corresponding digraphs achieving this upper bound. In particular, if \(D\) contains no odd directed cycles, then \(\gamma(D) \leq \frac{n}{2}\).

A graph is called degree-magic if it admits a labelling of the edges by integers \(\{1, 2, \ldots, |E(G)|\}\) such that the sum of the labels of the edges incident with any vertex \(v\) is equal to \(\left(1 + |E(G)|\right)/2 \deg(v)\). In this paper, we show that a class of join graphs are degree-magic.

A vertex-deleted unlabeled subgraph of a graph \(G\) is called a card of \(G\). A card of \(G\) with which the degree of the deleted vertex is also given is called a degree-associated card or dacard of \(G\). The degree-associated reconstruction number, \(\mathrm{drn}(G)\), of a graph \(G\) is the size of the smallest collection of dacards of \(G\) that uniquely determines \(G\). The maximal subgraph without end vertices of a graph \(G\) that is not a tree is called the pruned graph of \(G\). It is shown that \(\mathrm{drn}\) of some connected graphs with regular pruned graph is \(2\) or \(3\).

The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Feng et al. in [The hyper-Wiener index of bicyclic graphs, Utilitas Math., \(84(2011) 97-104\)] determined the bicyclic graphs having the largest Wiener index. In this article, we determine the graphs having the second up to seventh largest Wiener indices among all bicyclic graphs with \(n\) vertices.