Determining the biplanar crossing number of the graph \(C_n \times C_n \times C_n \times P_n\) was a problem proposed in a paper by Czabarka, Sykora, Székely, and Vito [2]. We find as a corollary to the main theorem of this paper that the biplanar crossing number of the aforementioned graph is zero. This result follows from the decomposition of \(C_n \times C_n \times C_n \times P_m\) into one copy of \(C_{n^2} \times P_{lm},l-2\) copies of \(C_{n^2} \times P_m\), and a copy of \(C_{n^2} \times P_{2m}\).

Let \(A_n\) be the alternating group of degree \(n\) with \(n \geq 5\). Set \(S = \{(1ij), (1ji) \mid 2 \leq i, j \leq n, i \neq j\}\). In this paper, it is shown that the full automorphism group of the Cayley graph \(\mathrm{Cay}(A_n, S)\) is the semi-product \(R(A_n) \rtimes \mathrm{Aut}(A_n, S)\), where \(R(A_n)\) is the right regular representation of \(A_n\) and \(\mathrm{Aut}(A_n, S) = \{\phi \in \mathrm{Aut}(A_n) \mid S^\phi = S\} \cong \mathrm{S_{n-1}}\).

Topological indices of graphs, and trees in particular, have been vigorously studied in the past decade due to their many applications in different fields. Among such indices, the number of subtrees (BC-subtrees), along with their variations, have received much attention. In this paper, we provide some new evaluation results related to these two indices on specific structures, such as generalized Bethe trees, Bethe trees, and dendrimers, which are of practical interest. Using generating functions, we also examine the asymptotic behavior of subtree (resp. BC-subtree) density of dendrimers.

For an integer \(k \geq 0\), a graphical property \(P\) is said to be \(k\)-stable if whenever \(G + uv\) has property \(P\) and \(d_G(u) + d_G(v) \geq k\), where \(uv \notin E(G)\), then \(G\) itself has property \(P\). In this note, we present spectral sufficient conditions for several stable properties of a graph.

Let \(G\) be a connected graph. The degree resistance distance of \(G\) is defined as \(D_R(G) = \sum\limits_{\{u,v\} \in V(G)} (d(u) + d(v))r(u,v)\), where \(d(u)\) (and \(d(v)\)) is the degree of the vertex \(u\) (and \(v\)), and \(r(u,v)\) is the resistance distance between vertices \(u\) and \(v\). A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than \(3\) in its unique cycle. In this paper, we determine the minimum and maximum degree resistance distance among all fully loaded unicyclic graphs with \(n\) vertices, and characterize the extremal graphs.

The cyclic edge-connectivity of a cyclically separable graph \(G\), denoted by \(c\lambda(G)\), is the minimum cardinality of all edge subsets \(F\) such that \(G – F\) is disconnected and at least two of its components contain cycles. Since \(c\lambda(G) \leq \zeta(G)\), where \(\zeta(G) = \min\{w(A) \mid A \text{ induces a shortest cycle in } G\}\), for any cyclically separable graph \(G\), a cyclically separable graph \(G\) is said to be cyclically optimal if \(c\lambda(G) = \zeta(G)\). The mixed Cayley graph is a kind of semi-regular graph. The cyclic edge-connectivity is a widely studied parameter, which can be used to measure the reliability of a network. Because previous work studied cyclically optimal mixed Cayley graphs with girth \(g \geq 5\), this paper focuses on mixed Cayley graphs with girth \(g < 5\) and gives some sufficient and necessary conditions for these graphs to be cyclically optimal.

Let \(p\) be an odd prime, \(q\) be a prime power coprime to \(p\), and \(n\) be a positive integer. For any positive integer \(d \leq n\), let \(g_1(x) = {x^{p^{n-d}} – 1}\),\(g_2(x)=1+{x^{p^{n – d+1}}}+x^{2p^{n-d+1}}+ \ldots +x^{(p^{d-1}-1)p^{n-d+1}}\),and , \(g_3(x) =1+x^{p^{n-d}}+x^{2p^{n-d}}+ \ldots +x^{(p-1)p^{n-d}} \). In this paper, we determine the weight distributions of \(q\)-ary cyclic codes of length \(pn\) generated by the polynomials \(g_1(x)\), \(g_2(x)\), \(g_3(x)\), \(g_4(x)\), and \(g_5(x)\), by employing the techniques developed in Sharma \& Bakshi [11]. Keywords: cyclic codes, Hamming weight, weight spectrum.

A construction of authentication codes with arbitration from singular symplectic geometry over finite fields is given, and the parameters of the codes are computed. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of success for different types of deceptions are also computed.

Let \(M\) be a simple connected binary matroid with corank at least two such that \(M\) has no connected hyperplane. Seymour proved that \(M\) has a non-trivial series class. We improve this result by proving that \(M\) has at least two disjoint non-trivial series classes \(L_1\) and \(L_2\) such that both \(M \backslash L_1\) and \(M \backslash L_2\) are connected. Our result extends the corresponding result of Kriesell regarding critically \(2\)-connected graphs.

For a non-complete graph \(\Gamma\), a vertex triple \((u,v,w)\) with \(v\) adjacent to both \(u\) and \(w\) is called a \(2\)-geodesic if \(u \neq w\) and \(u,w\) are not adjacent. Then \(\Gamma\) is said to be \(2\)-geodesic transitive if its automorphism group is transitive on both arcs and \(2\)-geodesics. In this paper, we classify the family of connected \(2\)-geodesic transitive graphs of valency \(3p\), where \(p\) is an odd prime.