Let \(K_r\) be the complete graph on \(r\) vertices in which there exists an edge between every pair of vertices, \(K_{m,n}\) be the complete bipartite graph with \(m\) vertices in one partition and \(n\) vertices in the other partition, where each vertex in one partition is adjacent to each vertex in the other partition, and \(K(n, r)\) be the complete \(r\)-partite graph \(K_{n,n,…,n}\) where each partition has \(n\) vertices. In this paper, we determine the minimum number of monochromatic stars \(K_{1,p}\), \( \forall p \geq 2\), in any \(t\)-coloring (\(t \geq 2\)) of edges of \(K_r\), \(K_{m,n}\), and \(K(n, r)\). Also, we prove that these lower bounds are sharp for all values of \(m, n, p, r\), and \(t\) by giving explicit constructions.

In this paper, we prove that if the toughness of a \(k\)-tree \(G\) is at least \(\frac{k+1}{3}\), then \(G\) is panconnected for \(k \geq 3\), or \(G\) is vertex pancyclic for \(k = 2\). This result improves a result of Broersma, Xiong, and Yoshimoto.

Since the Wiener index has been successful in the study of benzenoid systems and boiling points of alkanes, it is natural to examine this number for the study of fullerenes, most of whose cycles are hexagons. This topological index is equal to the sum of distances between all pairs of vertices of the respective graph. It was introduced in \(1947\) by one of the pioneers of this area, Harold Wiener, who realized that there are correlations between the boiling points of paraffins and the structure of the molecules. The present paper is the first attempt to compute the Wiener index of an infinite class of fullerenes. Further, we obtain a correlation between the values of the Wiener index and the boiling point of such fullerenes for the first time.

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.

We explore new combinatorial properties of overpartitions, which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition, and regular partition functions. We provide both generating function and bijective proofs. We also prove congruences for certain overpartition functions combinatorially.

Let \(G\) be a simple graph on \(n\) vertices. The Laplacian Estrada index of \(G\) is defined as \(LEE(G) = \sum_{i=1}^{n} e^{\mu_i}\), where \(\mu_1, \mu_2, \dots, \mu_n\) are the Laplacian eigenvalues of \(G\). In this paper, threshold graphs on \(n\) vertices and \(m\) edges having maximal and minimal Laplacian Estrada index are determined, respectively.

In this paper, formulas of the resistance distance for the arbitrary two-vertex resistance of \(G\), \(H = G_1 \boxdot G_2\) and \(G_1 \boxminus G_2\) in the electrical networks are obtained in a much simpler way. Furthermore, \(K_f(G_1 \boxdot G_2)\) and \(K_f(G_1 \boxminus G_2)\) can be expressed as a combination of \(K_f(G_1)\) and \(K_f(G_2)\).

Networks are important structures and appear in many different applications and settings. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Centrality parameters play an important role in the field of network analysis. Numerous studies have proposed and analyzed several centrality measures. These concepts measure the importance of a node’s position in a network. In this paper, vertex residual closeness \((VRC)\) and normalized vertex residual closeness \((NVRC)\) of some splitting networks modeled by splitting graphs are obtained.

Let \(T\) be an isosceles right triangle and let \(S_1, S_2, S_3, \dots\) be the homothetic copies of a square \(S\). In this paper, we consider the parallel covering and packing of \(T\) with the sequence \(\{S_n\}\) of squares.

A graph \(G\) is called super edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \rightarrow \{1, 2, \dots, |V(G)| + |E(G)|\}\) such that \(f(V(G)) = \{1, 2, \dots, |V(G)|\}\) and \(f(u) + f(v) + f(uv)\) is a constant for each \(uv \in E(G)\). The super edge-magic deficiency, \(\mu_s(G)\), of a graph \(G\) is defined as the smallest nonnegative integer \(n\) with the property that the graph \(G \cup nK_1\) is super edge-magic, or \(+\infty\) if there exists no such integer \(n\). In this paper, the super edge-magic deficiency of certain 2-regular graphs with two components is computed, which leads us to a conjecture on the super edge-magic deficiency of graphs in this class.

From a computer search, new minimum sizes for the maximal partial spreads in \(PG(3,q)\) have been obtained for \(q = 8, 9, 16\) and for every \(q\) such that \(25 \leq q \leq 101\). Furthermore, density results in the cases \(q = 8, 9, 16, 19, 23, 25, 27\) have been obtained. Finally, the already known exceptional size \(45\) for \(q = 7\) has been found again.

We decompose the complete multigraph \(K(v, \lambda)\) into copies of a graph \(H_i\) (\(i = 1, 2, 3\)). Each \(H_i\) is a near-triangle in that it is connected and has \(3\) vertices. In several cases, the decompositions are completed using classical combinatorial sequences due to Langford and Skolem.

It may be desired to seat \(n\) people along a row (as at a lunch counter), or \(n+1\) people around a circular table, in \(n\) consecutive rounds of seating, so that each person \(x\) has every other person \(y\) on their right exactly once, and on their left exactly once, in one of the seatings. Alternatively, it may be desired to seat \(2n\) people along a row, or \(2n + 1\) people around a circular table, in only \(n\) consecutive rounds, so that each person \(x\) is adjacent to every other person \(y\) (either on the right or the left) exactly once. We show that these problems are solved using the rows of Tuscan squares to specify the successive rounds of seatings.

Let \(\mathbb{F}_q^(n+1)\) denote the \((n+l)\)-dimensional projective space over a finite field \(\mathbb{F}_q\). For a fixed integer \(m \leq \min\{n,l\}\), denote by \(\mathcal{L}_o^m(\mathbb{F}_q^{n+1})\) the set of all subspaces of type \((t,t_1)\), where \(t_1 \leq t \leq m\). Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute all its parameters.

If \(X\) is a geodesic metric space and \(x_1, x_2, x_3 \in X\), a geodesic triangle \(T = \{x_1, x_2, x_3\}\) is the union of the three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) in \(X\). The space \(X\) is \(\delta\)-hyperbolic (in the Gromov sense) if any side of \(T\) is contained in a \(5\delta\)-neighborhood of the union of the two other sides, for every geodesic triangle \(T\) in \(X\). We denote by \(\delta(X)\) the sharp hyperbolicity constant of \(X\), i.e., \(\delta(X) := \inf\{\delta \geq 0: X \text{ is } \delta\text{-hyperbolic}\}\). The main result of this paper is the inequality \(\delta(G) \leq \delta(\mathcal{L}(G))\) for the line graph \(\mathcal{L}(G)\) of every graph \(G\). We prove also the upper bound \(\delta(L(G)) \leq 5\delta(G) + 3l_{\max}\), where \(\max\) is the supremum of the lengths of the edges of \(G\). Furthermore, if every edge of \(G\) has length \(k\), we obtain \(\delta(G) \leq \delta(\mathcal{L}(G)) \leq 5\delta(G) + 5k/2\).

For graphs \(G\) and \(H\), the size-balanced Ramsey multipartite number \(m_j(G, H)\) is defined as the smallest positive integer \(s\) such that any arbitrary red/blue coloring of the graph \(K_{s,s}\) forces the appearance of a red \(G\) or a blue \(H\). In the main case of this paper, we generalize methods used in finding bipartite Ramsey numbers for \(b(nK_2, mK_2)\) to finding the balanced Ramsey multipartite number \(m_j(nK_2, mK_2)\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-vertex corona of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(|V(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(V(G_1)\) to every vertex in the \(i\)th copy of \(G_2\). The subdivision-edge corona of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(|I(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(I(G_1)\) to every vertex in the \(i\)th copy of \(G_2\), where \(I(G_1)\) is the set of inserted vertices of \(S(G_1)\). In this paper, we determine the generalized characteristic polynomial of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)). As applications, the results on the spectra of \( G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) enable us to construct infinitely many pairs of \(\Phi\)-cospectral graphs. The adjacency spectra of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) help us to construct many infinite families of integral graphs. By using the Laplacian spectra, we also obtain the number of spanning trees and Kirchhoff index of \(G_1 \odot G_2\) and \(G_1 \ominus G_2\), respectively.

In this paper, we study arc-transitive pentavalent graphs of order \(4p^n\), where \(p\) is a prime and \(n\) is a positive integer. It is proved that no such graph exists for each prime \(p \geq 5\), and all such graphs with \(p = 2\) or \(3\) which are \(G\)-basic (that is, \(G\) has no non-trivial normal subgroup such that the graph is a normal cover of the corresponding normal quotient graph) are determined. Moreover, as an application, arc-transitive pentavalent graphs of order \(4p^2\) and \(4p^3\) are determined.

In \(1982\), Beutelspacher and Brestovansky determined the \(2\)-color Rado number of the equation \[x_1+ x_2 x + \ldots +x_{m-1} =x_{ m} \] for all \(m \geq 3\). Here we extend their result by determining the 2-color Rado number of the equation \[x_1 +x_2 + \dots + x_n = y_1 +y_2+ \ldots + y_k\] for all \(n \geq 2\) and \(k \geq 2\). As a consequence, we determine the 2-color Rado number of \[x_1+ x_2 + \ldots + x_n = a_1 y_1 + \dots + a_\ell y_\ell\] in all cases where \(n \geq 2\) and \(n \geq a_1 + \dots + a_\ell\), and in most cases where \(n \geq 2\) and \(2n \geq a_1 + \dots + a_\ell\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). The set of inserted vertices of \(S(G)\) is denoted by \(I(G)\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-edge-vertex join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(V(G_2)\). The subdivision-edge-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(I(G_2)\). The subdivision-vertex-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(V(G_1)\) to every vertex in \(I(G_2)\). In this paper, we obtain the formulas for resistance distance of \(G_1 \odot G_2\), \(G_1 \ominus G_2\), and \(G_1 \odot G_2\).

A hypergraph is intersecting if any two different edges have exactly one common vertex, and an \(n\)-quasicluster is an intersecting hypergraph with \(n\) edges, each one containing at most \(n\) vertices, and every vertex is contained in at least two edges. The Erdős-Faber-Lovász Conjecture states that the chromatic number of any \(n\)-quasicluster is at most \(n\). In the present note, we prove the correctness of the conjecture for a new infinite class of \(n\)-quasiclusters using a specific edge coloring of the complete graph.

Let \(G\) be a graph of order \(n\) and let \(\Phi(G, \lambda) = \det(\lambda I_n – L(G)) = \sum_{k=0}^{n}(-1)^k c_k(G) \lambda^{n-k}\) be the characteristic polynomial of the Laplacian matrix of a graph \(G\). In this paper, we identify the minimal Laplacian coefficients of unicyclic graphs with \(n\) vertices and diameter \(d\). Finally, we characterize the graphs with the smallest and the second smallest Laplacian-like energy among the unicyclic graphs with \(n\) vertices and fixed diameter \(d\).

The balanced hypercube, which is a variant of the hypercube, is proposed as a novel inter-processor network. Among the attractive properties of the balanced hypercube, the most special one is that each processor has a backup processor sharing the same neighborhood. A connected graph \(G\) with at least \(2m + 2\) vertices is said to be \(m\)-extendable if it possesses a matching of size \(m\) and every such matching can be extended to a perfect matching of \(G\). In this paper, we prove that the balanced hypercube \(BH_n\) is \(m\)-extendable for every \(m\) with \(1 \leq m \leq 2n – 2\), and our result is optimal.

A set \(W \subseteq V(G)\) is called a resolving set, if for each two distinct vertices \(u, v \in V(G)\) there exists \(w \in W\) such that \(d(u, w) \neq d(v, w)\), where \(d(x, y)\) is the distance between the vertices \(x\) and \(y\). A resolving set for \(G\) with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a unique basis graph. In this paper, we study some properties of unique basis graphs.

In this paper, we study the number of 1-factors and edge-colorings of the Möbius ladder graphs. We find exact formulae for such numbers and show that there are exponentially many 1-factors and edge-colorings in such graphs. As applications, we show that every “man-made” triangular embedding for \(K_{12m+7}\), by combining the current graphs with those of Youngs and Ringel, permits exponentially many “Grünbaum colorings” (i.e., 3-edge-colored triangulations in such a way that each triangle receives three distinct colors).

Multi-receiver authentication codes with dynamic sender (\(DMRA\)-codes) are extensions of traditional group communication systems in which any member of a group can broadcast an authenticated message such that all other group members can individually verify its authenticity, and some malicious participants of the group cannot successfully impersonate the potential sender, or substitute a transmitted message. In this paper, a construction of \(DMRA\)-code will be given using linear code and its unconditional security is also guaranteed.

We consider the problem of finding quasiperiodicities in Fibonacci strings. A factor \(u\) of a string \(y\) is a cover of \(y\) if every letter of \(y\) falls within some occurrence of \(u\) in \(y\). A string \(v\) is a seed of \(y\) if it is a cover of a superstring of \(y\). A left seed of a string \(y\) is a prefix of \(y\) that is a cover of a superstring of \(y\). Similarly, a right seed of a string \(y\) is a suffix of \(y\) that is a cover of a superstring of \(y\). In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings; we identify all covers, left/right seeds, and seeds of a Fibonacci string and all covers of a circular Fibonacci string.

We investigate a modifications of the well-known irregularity strength of graphs, namely the total edge irregularity strength and the total vertex irregularity strength. In this paper, we determine the exact value of the total edge (vertex) irregularity strength for convex polytope graphs with pendent edges.

Two \(G\)-designs \((X, \mathcal{A}_1)\) and \((X, \mathcal{A}_2)\) are said to intersect in \(m\) blocks if \(|\mathcal{A}_1 \cap \mathcal{A}_2| = m\). In this paper, we complete the solution of the intersection problem for \(G\)-designs, where \(G\) is a connected graph of size five which contains a cycle.

In this paper we discuss how the addition of a new edge affects the total edge irregularity strength of a graph.

Let \(G\) be a connected graph and \(k > 1\) be an integer. The local \(k\)-restricted edge connectivity \(\lambda_k(X,Y)\) of \(X,Y\) in \(G\) is the maximum number of edge-disjoint \(X\)-\(Y\) paths for \(X,Y \subseteq V\) with \(|X| = |Y| = k\), \(X \cap Y = \emptyset\), \(G[X]\) and \(G[Y]\) are connected. The \(k\)-restricted edge connectivity of \(G\) is defined as \(\lambda_k(G) = \min\{\lambda_k(X,Y) : X,Y \subseteq V, |X| = |Y| = k, X \cap Y = \emptyset, G[X] \text{ and } G[Y]\) are connected. Then \(G\) is local optimal \(k\)-restricted edge connected if \(\lambda_k(X,Y) = \min\{w(X), w(Y)\}\) for all \(X,Y \subseteq V\) with \(|X| = |Y| = k\), \(G[X]\) and \(G[Y]\) are connected, where \(w(X) = |E(X, \overline{X})|\). If \(\lambda_k(G) = \xi_k(G)\), where \(\xi_k(G) = \min\{w(X) : U \subset V, |U| = k \text{ and } G[U] \text{ is connected}\}\), then \(G\) is called \(\lambda_k\)-optimal. In this paper, we obtain several sufficient conditions for a graph to be \(3\)-optimal (or local optimal \(k\)-restricted edge connected).

A graph \(G\) is called edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \to \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv)\) is a constant for each \(uv \in E(G)\). Also, \(G\) is called super edge-magic if \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\). Moreover, the super edge-magic deficiency, \(\mu_s(G)\), of a graph \(G\) is defined to be the smallest nonnegative integer \(n\) with the property that the graph \(G \cup nK_1\) is super edge-magic, or \(+\infty\) if there exists no such integer \(n\). In this paper, we introduce the notion of the sequential number, \(\sigma(G)\), of a graph \(G\) without isolated vertices to be either the smallest positive integer \(n\) for which it is possible to label the vertices of \(G\) with distinct elements from the set \(\{0, 1, \ldots, n\}\) in such a way that each \(uv \in E(G)\) is labeled \(f(u) + f(v)\) and the resulting edge labels are \(|E(G)|\) consecutive integers, or \(+\infty\) if there exists no such integer \(n\). We prove that \(\sigma(G) = \mu_s(G) + |V(G)| – 1\) for any graph \(G\) without isolated vertices, and \(\sigma(K_{m,n}) = mn\) for every two positive integers \(m\) and \(n\), which allows us to settle the conjecture that \(\mu_s(K_{m,n}) = (m-1)(n-1)\) for every two positive integers \(m\) and \(n\).

Let \(G = (V, E)\) be a graph. An edge labeling \(f: E \to \mathbb{Z}_2\) induces a vertex labeling \(f^*: V \to \mathbb{Z}_2\) defined by \(f^*(v) = \sum_{uv \in E} f(uv) \pmod{2}\). For each \(i \in \mathbb{Z}_2\), define \(E_i(f) = |f^{-1}(i)|\) and \(V_i(f) = |(f^*)^{-1}(i)|\). We call \(f\) edge-friendly if \(|E_1(f) – E_0(f)| \leq 1\). The edge-friendly index \(I_f(G)\) is defined as \(V_1(f) – V_0(f)\), and the full edge-friendly index set \(FEFI(G)\) is defined as \(\{I_f(G): f \text{ is an edge-friendly labeling}\}\). Further, the edge-friendly index set \(EFI(G)\) is defined as \(\{|I_f(G)|: f \text{ is an edge-friendly labeling}\}\). In this paper, we study the full edge-friendly index set of the star \(K_{1,n}\), \(2\)-regular graph, wheel \(W_n\), and \(m\) copies of path \(mP_n\), \(m \geq 1\).

An acyclic total coloring is a proper total coloring of a graph \(G\) such that there are at least \(4\) colors on vertices and edges incident with a cycle of \(G\). The acyclic total chromatic number of \(G\), \(\chi”_a(G)\), is the least number of colors in an acyclic total coloring of \(G\). In this paper, we prove that for every plane graph \(G\) with maximum degree \(\Delta\) and girth \(g(G)\), \(\chi_a(G) = \Delta+1\) if (1) \(\Delta \geq 9\) and \(g(G) \geq 4\); (2) \(\Delta \geq 6\) and \(g(G) \geq 5\); (3) \(\Delta \geq 4\) and \(g(G) \geq 6\); (4) \(\Delta \geq 3\) and \(g(G) \geq 14\).

Codes in \(l_{p\gamma}\)-spaces, introduced by the author in [3], are a natural generalization of one-dimensional codes in \(RT\)-spaces [6] to block coding and have applications in different areas of combinatorial/discrete mathematics, e.g., in the theory of uniform distribution, experimental designs, cryptography, etc. In this paper, we introduce various types of weight enumerators in \(l_{p\gamma}\)-codes, viz., exact weight enumerator, complete weight enumerator, block weight enumerator, and \(\gamma\)-weight enumerator. We obtain the MacWilliams duality relation for the exact and complete weight enumerators of an \(l_{p\gamma}\)-code.

We introduce a theorem on bipartite graphs, and some theorems on chains of two and three complete graphs, considering when they are combination or non-combination graphs, present some families of combination graphs. We give a survey for trees of order \(\leq 10\), which are all combination graphs.

A set of vertices in a graph \(G\) without isolated vertices is a total dominating set (TDS) of \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\). The minimum cardinality of a TDS of \(G\) is the total domination number \(\gamma_t(G)\) of \(G\). In this paper, the total domination number of generalized \(n\)-graphs and \(m \times n\) ladder graphs is determined.

We identify a graph without proper cycles, which is comatching with a cycle,The result is then extended to certain general families of graphs with cyclomatic number \(1\), formed by attaching trees to cycles.

A vertex \(v \in V(G)\) is said to be a self vertex switching of \(G\) if \(G\) is isomorphic to \(G^v\), where \(G^v\) is the graph obtained from \(G\) by deleting all edges of \(G\) incident to \(v\) and adding all edges incident to \(v\) which are not in \(G\). In [6], the author characterized connected unicyclic graphs each with a self vertex switching. In this paper, we characterize disconnected unicyclic graphs each with a self vertex switching.

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. A multi-wheel graph is a graph obtained from \(s\) cycles \(C_{n_1}, C_{n_2}, \ldots, C_{n_s}\) (\(s \geq 1\)) by adding a new vertex, say \(w\), and edges joining \(w\) to all the vertices of the \(s\) cycles. In this article, we solve a conjecture posed by Yu et al. in 2006 and prove that it is not always true. Furthermore, the classification problem of multi-wheel graphs on \(f\)-colorings is solved completely.

For a connected graph \(G = (V, E)\) of order at least two, a chord of a path \(P\) is an edge joining two non-adjacent vertices of \(P\). A path \(P\) is called a monophonic path if it is a chordless path. A longest \(x\)-\(y\) monophonic path is called an \(x\)-\(y\) detour monophonic path. A set \(S\) of vertices of \(G\) is a detour monophonic set of \(G\) if each vertex \(v\) of \(G\) lies on an \(x\)-\(y\) detour monophonic path for some \(x\) and \(y\) in \(S\). The minimum cardinality of a detour monophonic set of \(G\) is the detour monophonic number of \(G\) and is denoted by \(dm(G)\). For any two vertices \(u\) and \(v\) in \(G\), the monophonic distance \(dm(u,v)\) from \(u\) to \(v\) is defined as the length of a \(u\)-\(v\) detour monophonic path in \(G\). The monophonic eccentricity \(em(v)\) of a vertex \(v\) in \(G\) is the maximum monophonic distance from \(v\) to a vertex of \(G\). The monophonic radius \(rad_{m}(G)\) of \(G\) is the minimum monophonic eccentricity among the vertices of \(G\), while the monophonic diameter \(diam_{m}(G)\) of \(G\) is the maximum monophonic eccentricity among the vertices of \(G\). It is shown that for positive integers \(r\), \(d\), and \(n \geq 4\) with \(r < d\), there exists a connected graph \(G\) with \(rad_{m}(G) = r\), \(diam_{m}(G) = d\), and \(dm(G) = n\). Also, if \(p\), \(d\), and \(n\) are integers with \(2 \leq n \leq p-d+4\) and \(d \geq 3\), there is a connected graph \(G\) of order \(p\), monophonic diameter \(d\), and detour monophonic number \(n\). Further, we study how the detour monophonic number of a graph is affected by adding some pendant edges to the graph.