In Algebraic Graph Theory, Biggs \([2]\) gives a method for finding the chromatic polynomial of any connected graph by computing the Tutte polynomial. It is used by Biggs \([2]\) to compute the chromatic polynomial of Peterson’s graph. In \(1972\) Sands \([4]\) developed a computer algorithm using matrix operations on the incidence matrix to compute the Tutte Polynomial. In \([1]\), Anthony finds the worst-case time-complexity of computing the Tutte Polynomial. This paper shows a method using group-theoretical properties to compute the Tutte polynomial for Cayley graphs which improves the time-complexity.

Let \(N({Z})\) denote the set of all positive integers (integers). The sum graph \(G_S\) of a finite subset \(S \subset N({Z})\) is the graph \((S, E)\) with \(uv \in E\) if and only if \(u+v \in S\). A graph \(G\) is said to be an (integral) sum graph if it is isomorphic to the sum graph of some \(S \subset N({Z})\). The (integral) sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which when added to \(G\) result in an (integral) sum graph. A mod (integral) sum graph is a sum graph with \(S \subset {Z}_m \setminus \{0\}\) (\(S \subset {Z}_m\)) and all arithmetic performed modulo \(m\) where \(m \geq |S|+1\) (\(m \geq |S|\)). The mod (integral) sum number \(\rho(G)\) of \(G\) is the least number \(\rho\) (\(\psi\)) of isolated vertices \(\rho K_1\) (\(\psi K_1\)) such that \(G \cup \rho K_1\) (\(G \cup \psi K_1\)) is a mod (integral) sum graph. In this paper, the mod (integral) sum numbers of \(K_{r,s}\) and \(K_n – E(K_r)\) are investigated and bounded, and \(n\)-spoked wheel \(W_n\) is shown to be a mod integral sum graph.

In this paper, the forcing domination numbers of the graphs \(P_n \times P_3\) and \(C_n \times P_3\) are completely determined. This improves the previous results on the forcing domination numbers of \(P_n \times P_2\) and \(C_n \times P_2\).

The stabilizers of the minimum-weight codewords of the binary codes obtained from the strongly regular graphs \(T(n)\) defined by the primitive rank-\(3\) action of the alternating groups \(A_n\), where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\) are examined. For a codeword \(w\) of minimum-weight in the binary code \(C\) obtained as stated above, from an adjacency matrix of the triangular graph \(T(n)\) defined by the primitive rank-3 action of the alternating groups \(A_n\) where \(n \geq 5\), on \(\Omega^{(2)}\), the set of duads of \(\Omega = \{1,2,\ldots,n\}\), we determine the stabilizer \(Aut(C)_w\) in \(Aut(C)\) and show that \(Aut(C)_w\) is a maximal subgroup of \(Aut(C)\).

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of strong orientations of \(G\). Define \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\) and \(\rho(G) = \overrightarrow{d}(G) – d(G)\), where \(d(D)\) (resp. \(d(G)\)) denotes the diameter of the digraph \(D\) (resp. graph \(G\)). In this paper, we determine the exact value of \(\rho(K_r \times K_s)\) for \(r \leq s\) and \((r,s) \not\in \{(3,5), (3,6), (4,4)\}\), where \(K_r \times K_s\) denotes the tensor product of \(K_r\) and \(K_s\). Using the results obtained here, a known result on \(\rho(G)\), where \(G\) is a regular complete multipartite graph is deduced as corollary.