A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order \(40p\) for each prime \(p\). It is shown that a connected pentavalent symmetric graph of order \(40p\) exists if and only if \(p = 3\), and up to isomorphism, there are only two such graphs.

A broadcast on a graph \(G\) is a function \(f: V \to \{0, \dots, diam(G)\}\) such that for every vertex \(v \in V(G)\), \(f(v) \leq e(v)\), where \(diam(G)\) denotes the diameter of \(G\) and \(e(v)\) denotes the eccentricity of vertex \(v\). The upper broadcast domination number of a graph is the maximum value of \(\sum_{v \in V} f(v)\) among all minimal broadcasts \(f\) for which each vertex of the graph is within distance \(f(v)\) from some vertex \(v\) having \(f(v) \geq 1\). We give a new upper bound on the upper broadcast domination number which improves a previous result of Dunbar et al. in [Broadcasts in graphs, Discrete Applied Mathematics 154 (2006) 59-75]. We also prove that the upper broadcast domination number of any grid graph \(G_{m,n} = P_m \Box P_n\) equals \(m(n – 1)\).

For a vertex \(v\) of a graph \(G\), Zhu, Li, and Deng introduced the concept of implicit degree \(id(v)\), according to the degrees of the neighbors of \(v\) and the vertices at distance \(2\) with \(v\) in \(G\). For a subset \(S \subseteq V(G)\), let \(i\Delta_2(G, S)\) denote the maximum value of the implicit degree sum of two vertices of \(S\). In this paper, we will prove: Let \(G\) be a \(2\)-connected graph on \(n \geq 3\) vertices and \(d\) be a nonnegative integer. If \(i\Delta_2(G, S) \geq d\) for each independent set \(S\) of order \(\kappa(G) + 1\), then \(G\) has a cycle of length at least \(\min\{d, n\}\).

For a nonempty graph \(G = (V(G), E(G))\), a signed cycle dominating function on \(G\) is introduced by Xu in 2009 as a function \(f : E(G) \to \{1, -1\}\) such that \(\sum_{e \in E(C)} f(e) \geq 1\) for any induced cycle \(C\) of \(G\). A set \(\{f_1, f_2, \dots, f_d\}\) of distinct signed cycle dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(e) \leq 1\) for each \(e \in E(G)\), is called a signed cycle dominating family (of functions) on \(G\). The maximum number of functions in a signed cycle dominating family on \(G\) is the signed cycle domatic number of \(G\), denoted by \(d’_{sc}(G)\). In this paper, we study the signed cycle domatic numbers in graphs and present sharp bounds for \(d’_{sc}(G)\). In addition, we determine the signed cycle domatic number of some special graphs.

Using partition theoretic methods we combinatorially interpret the four Ae Rogers—Ramanujan identities of Andrews, Schilling and Wamaar.

Let \(p > 165\) be a prime and let \(G\) be a cyclic group of order \(p\). Let \(S\) be a minimal zero-sum sequence with elements over \(G\), i.e., the sum of elements in \(S\) is zero, but no proper nontrivial subsequence of \(S\) has sum zero. We call \(S\) unsplittable, if there do not exist \(g \in S\) and \(x, y \in G\) such that \(g = x + y\) and \(Sg^{-1}x y\) is also a minimal zero-sum sequence. In this paper, we determine the structure of \(S\) which is an unsplittable minimal zero-sum sequence of length \(\frac{p-1}{2}\) or \(\frac{p-3}{2}\). Furthermore, if \(S\) is a minimal zero-sum sequence with \(|S| \geq \frac{p-3}{2}\), then \(ind(S) \leq 2\).

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1, G_2)\) is the smallest integer \(x\) such that for any graph \(G\) of order \(n\), either \(G\) contains \(G_1\) or the complement of \(G\) contains \(G_2\). In this paper, we study a large class of trees \(T\) as studied by Cockayne in [3], including paths and trees which have a vertex of degree one adjacent to a vertex of degree two, as special cases. We evaluate some \(R(T’_m, B_m)\), where \(T’_n \in \mathbb{T}\) and \(B_m\) is a book of order \(m+2\). Besides, some bounds for \(R(T’_n, B_n)\) are obtained.

Graceful labeling of graphs is used in radar codes. In this work, we introduce a new version of gracefulness, which we call edge-even graceful labeling of graphs. We establish a necessary and sufficient condition for edge-even graceful labeling of path graphs \(P_n\), cycle graphs \(C_n\), and star graphs \(K_{1,n}\). We also prove some necessary and sufficient conditions for some path and cycle-related graphs, namely, Friendship, Wheel, Double wheel, and Fan graphs.

The Hamiltonian problem is a classical problem in graph theory. Most of the research on the Hamiltonian problem is looking for sufficient conditions for a graph to be Hamiltonian. For a vertex \(v\) of a graph \(G\), Zhu, Li, and Deng introduced the concept of implicit degree \(id(v)\), according to the degrees of its neighbors and the vertices at distance \(2\) with \(v\) in \(G\). In this paper, we will prove that: Let \(G\) be a \(2\)-connected graph on \(n \geq 3\) vertices. If the maximum value of the implicit degree sums of \(2\) vertices in \(S\) is more than or equal to \(n\) for each independent set \(S\) with \(\kappa(G) + 1\) vertices, then \(G\) is Hamiltonian.

Let \((d_1, d_2, \dots, d_n)\) be a sequence of positive integers with \(n-1 \geq d_1 \geq d_2 \geq \dots \geq d_n\). We give a characterization of \((d_1, d_2, \dots, d_n)\) that is the degree sequence of a graph with cyclomatic number \(k\). This simplifies the characterization of Erdős-Gallai.