Let G be a connected (molecular) graph with the vertex set $V\left(G\right)=\{v_1,\cdots ,v_n\}$, and let $d_i$ and ${\sigma }_i$ denote, respectively, the vertex degree and the transmission of $v_i$, for $1\le i\le n$. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the $n\times n$ matrix whose $(i,j)$-entry is equal to $\frac{{\sigma }_i}{2d_i}+\frac{{\sigma }_j}{2d_j}$ if $v_i$ and $v_j$ are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

Let G be a simple and finite graph. A graph is said to be decomposed into subgraphs H₁ and H₂ which is denoted by G = H₁ ⊕ H₂, if G is the edge disjoint union of H₁ and H₂. If G = H₁ ⊕ H₂ ⊕ H₃ ⊕ ... ⊕ H_k, where H₁, H₂, H₃, ..., H_k are all isomorphic to H, then G is said to be H-decomposable. Futhermore, if H is a cycle of length m then we say that G is C_m-decomposable and this can be written as C_m|G. Where G × H denotes the tensor product of graphs G and H, in this paper, we prove the necessary and sufficient conditions for the existence of C₄-decomposition of K_m × K_n. Using these conditions it can be shown that every even regular complete multipartite graph G is C₄-decomposable if the number of edges of G is divisible by 4.

In this paper, we consider C₁₀-decompositions of the tensor product of complete graphs, K_m × K_n, m, n ≥ 3. It is proved that the necessary conditions for such a decomposition are also sufficient. The cycle C₁₀ decomposes K_m × K_n if and only if either (1) n ≡ 0 or 6 (mod 10) and m is odd, (2) m ≡ 0 or 6 (mod 10) and n is odd, or (3) m or n ≡ 1 or 5 (mod 10). Using these conditions it can be shown that every even regular complete multipartite graph G is C₁₀-decomposable if the number of edges of G is divisible by 10.

Let G be a simple, finite and connected graph. An outer-connected vertex edge dominating set, abbreviated OCVEDS of G is a subset D of vertices of G such that D is a vertex edge dominating set and the graph G\D is connected. The outer-connected vertex edge domination number of G is a minimum cardinality of a OCVEDS of G and is denoted by . This paper focuses on the OCVEDS in Cartesian product of graphs.

Given a graph $G$, a total labeling on $G$ is called edge-antimagic total (respectively, vertex-antimagic total) if all edge-weights (respectively, vertex-weights) are pairwise distinct. If a labeling on $G$ is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graph.