A family of graphs that are DGS but not DS

Cover numbers by certain graph families

An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps

We give a construction of a new family of Deza graphs of girth at least 5 that possess an arc-transitive group of automorphisms isomorphic to a Suzuki simple group \(Sz(q)\). To study their combinatorical properties, we elaborate some group-theoretic arguments involving classical results on the groups of given type.

The Hamming matrix of a graph arises from the notion of Hamming distance and provides a matrix-based framework for studying vertex dissimilarity. The corresponding Hamming energy, defined as the sum of the absolute values of the eigenvalues of the Hamming matrix, represents a natural spectral invariant that is closely related to the classical graph energy. In this paper, we investigate the Hamming matrix of two important families of graphs, namely sunlet graphs and barbell graphs. By applying the technique of equitable vertex partitions and methods from matrix spectral theory, we obtain explicit expressions for the H-spectrum and the H-energy of these graphs. Our results extend and complement existing studies on energy-like invariants for special classes of graphs.

Computation of Total Kulli-Basava Indices on a Few Specialized Families of Graphs

In this work, we study Laplacian spectrum of specific graph families constructed by applying join ([Formula: see text]) and direct product (×) operators on the complete graph ([Formula: see text]), complete bipartite graph ([Formula: see text]), hypercube graph ([Formula: see text]) and cycle graph ([Formula: see text]) for some integer [Formula: see text]. In particular, we derive complete Laplacian spectrum of the following graphs explicitly and determine for which [Formula: see text] they can be calculated: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text], (iv) [Formula: see text]. We note that the graphs [Formula: see text] are the regular graphs whose Laplacian spectrum are known. By applying graph operations on these graphs, we extend the existing knowledge of Laplacian spectra. Our results suggest that further exploration of the Laplacian spectra of other graph families, using similar graph operations, could yield valuable insights.

On an infinite family of integral and distance integral Cayley graphs with few distance eigenvalues

In computer science and graph theory, prism and antiprism graphs are crucial for network modeling, optimization, and network connectivity comprehension. Applications such as social network analysis, fault-tolerant circuit design, and parallel and distributed computing all make use of them. Their structured nature makes them important, since it offers a framework for researching intricate characteristics, including resilient design, communication patterns, and network efficiency. This work uses the electrically equivalent transformations technique to compute the explicit formulas for the number of spanning trees of three novel families of graphs that have been produced using triangular prisms with their distinctive iteration feature. Additionally, the relationship between these graphs’ average degree and entropy is examined and contrasted with the entropy of additional graphs that share the same average degree as these previously studied graphs.

Topological graph indices are the most important applications of graph theory. In the last few decades, such applications have been increasingly in use for mathematicians, chemists, pharmacologists. Due to these applications, a large number of these indices are defined and used in several areas. The first and second reduced neighborhood Zagreb indices have been recently defined by [Formula: see text] In this paper, following the recent study of reduced neighborhood Zagreb indices, we introduced the forgotten reduced neighborhood topological index, obtained values of some families of graphs and also computed the exact values of this index for graphene and honeycomb networks.

Let Aex(G) be the extended adjacency matrix of G. The eigenvalues of Aex(G) are called extended adjacency eigenvalues of G. The sum of the absolute values of eigenvalues of the Aex-matrix is called the extended adjacency energy Eex(G) of G. In this paper, we obtain the Aex-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the Aex-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the Aex-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the Aex-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the Aex-eigenvalues of the power graph of finite abelian groups.

Let (𝐺) be the eccentricity matrix of a graph 𝐺, and Spec(𝜀(𝐺)) be the eccentricity spectrum of 𝐺. Let [𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ] be the 𝐻-join of graphs 𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 and let 𝐻[𝐺] be lexicographic product of 𝐻 and 𝐺. This paper determines the eccentricity matrix of a 𝐻-join of graphs. Using this result, we obtain the following results. (i) We construct a family of 𝜀-cospectral graphs, (ii) We derive Spec(𝜀(𝐻[𝐺])) in terms of Spec(𝜀(𝐻)) if rad(𝐻) ≥ 3, where rad(𝐻) is the radius of 𝐺, (iii) We find Spec(𝜀(𝐾 𝑘 [𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ])) if Δ(𝐺 𝑖 ) ≤ |𝑉 (𝐺 𝑖 )| − 2 which generalizes some of the results from [Mahato et al. in Discrete Appl Math. (2020) 285:252–260], (iv) We determine Spec(𝜀(𝐻[𝐺 1 , 𝐺 2 , … , 𝐺 𝑘 ])) if rad(𝐻) ≥ 2 and 𝐺 𝑖 is complete whenever 𝑒 𝐻 (𝑖) = 2, which generalizes some of the results from [Mahato et al. in Discrete Appl Math. (2020) 285:252–260; Wang et al. in Discrete Math. (2019) 342(9):2636–2646]. Finally, we find the characteristic polynomial of (𝐾 1,𝑚 [𝐺 0 , 𝐺 1 , … , 𝐺 𝑚 ]) if 𝐺 𝑖 ’s are regular. As an application, we deduce some of the results from [Li et al. in Discrete Appl. Math. (2023) 336:47–55; Mahato et al. in Discrete Appl Math. (2020) 285:252–260; Patel et al. in Discrete Math. (2021) 344:112591 and Wang et al. (2018) Discrete Appl Math. 251:299–309].

Borodin-Kostochka Conjecture for a Family of P6-free Graphs

In this paper, we study the detour eccentric sum index (DESI) to obtain the Quantitative Structure–Property Relationship (QSPR) for different molecular structures. We establish theoretical bounds for this index and compute its values across fundamental graph families. Through correlation analyses between the physicochemical properties of molecular structures representing anti-malarial and breast cancer drugs, we show the high predictive value of two topological parameters, detour diameter (DD) and detour radius (DR). Specifically, DR shows strong positive correlations with boiling point, enthalpy, and flash point (up to 0.94), while DD is highly correlated with properties such as molar volume, molar refraction, and polarizability (up to 0.97). The DESI was then selected for detailed curvilinear regression modeling and comparison against the established eccentric distance sum index. For anti-malarial drugs, the second-order model yields the best fit. The DESI provides optimal prediction for boiling point, enthalpy, and flash point. In breast cancer drugs, the second-order model is again favored for properties except for melting point, best described by a third-order model. The results highlight how well the index captures subtle structural characteristics.

Let $G = (V, E)$ be a simple graph, and let $f: V \to \{0, 1, 2, 3\}$ be a function. A vertex \( u \) is considered an undefended vertex with respect to \( f \) if \( f(u) = 0 \) and there is no adjacent vertex \( v \) satisfying \( f(v) \geq 2 \). A function \( f \) is termed a generous Roman dominating function (GRD-function) if, for every vertex \( u \) with \( f(u) = 0 \), there exists at least one adjacent vertex \( v \) such that \( f(v) \geq 2 \) and the modified function \( f': V \to \{0,1,2,3\} \), defined as\( f'(u) = \alpha, \quad f'(v) = f(v) - \alpha,\) where \( \alpha \in \{1,2\} \), and\(f'(w) = f(w) \quad \text{for all } w \in V \setminus \{u, v\},\) ensures that no vertex remains undefended. The weight of a GRD-function \( f \) is defined as\(f(V) = \sum_{u \in V} f(u).\) The smallest possible weight of a GRD-function on \( G \) is known as the generous Roman domination number of \( G \), denoted by \( \gamma_{gR}(G) \). The generous Roman domination subdivision number, represented as \( \mathrm{sd}_{\gamma_{gR}}(G) \), is the minimum number of edges that must be subdivided (where each edge in \( G \) can be subdivided at most once) to increase the generous Roman domination number. In this paper, we establish upper bounds on the generous Roman domination subdivision number. Furthermore, we determine the exact value of this parameter for certain families of graphs, including paths, cycles, and ladders. Further, we present several sufficient conditions for a graph \( G \) to have a small value of \( sd_{\gamma_{gR}}(G) \).\end{abstract}\keywords{generous Roman domination, generous Roman domination subdivision number}

Given a simple undirected graph $G=(V(G),E(G))$, a vertex $u\in V(G)$ vertex-edge dominates the edge $xy\in E(G)$ if one of the following holds: $(1)$ $u=x$ or $u=y$, $(2)$ $ux\in E(G)$ or $uy\in E(G)$. A subset $S\subseteq V(G)$ is a vertex-edge dominating set of $G$ if for each $xy\in E(G)$, there exists $u\in S$ such that $u$ vertex-edge dominates $xy$. A vertex-edge dominating set $S\subseteq V(G)$ is a total vertex-edge dominating set if for each $u\in S$, there exists $v\in S$ for which $uv\in E(G)$. The minimum cardinality of a vertex-edge (resp. total vertex-edge) dominating set of $G$ is the vertex-edge domination number (resp. total vertex-edge domination number) of $G$. This paper investigates the vertex-edge domination and total vertex-edge domination in the join, corona, lexicographic product, complementary prism and edge corona of graphs. It provides complete characterizations of both the vertex-edge dominating sets and total vertex-edge dominating sets in these families of graphs, and establishes sharp bounds, if not the exact values, for their respective vertex-edge domination and total vertex-edge domination numbers.

A graph $G$ with $p$ vertices and $q$ edges is said to be edge-graceful if its edges can be labeled from $1$ through $q$, in such a way that the labels induced on the vertices by adding over the labels of incident edges modulo $p$ are distinct. Lo's Theorem is a known result under this topic, which states that if a graph $G$ with $p$ vertices and $q$ edges is edge-graceful, then $p\Big|\Big(q^{2}+q-\frac{p(p-1)}{2}\Big)$. Since the introduction of the concept of edge-graceful labeling, several works on the edge-graceful labeling of specific families of graphs have been conducted. This paper recalls the definition of water wheel graphs $(WW_n)$ and introduces two new related graphs, the right water wheel graphs $(RWW_n)$ and left water wheel graphs $(LWW_n)$. Next, using Lo's Theorem, and the concepts of divisibility and Diophantine equations, we proved the non-edge-gracefulness of these graphs. Then, using the same concepts, we determine all the edge-graceful splitting graphs of $WW_n$, $RWW_n$, and $LWW_n$, denoted as $S(WW_n)$, $S(RWW_n)$, and $S(LWW_n)$, respectively. Finally, Python computer programs were created and used to conclude that among these graph families, only $S(WW_3)$, $S(RWW_3)$, and $S(LWW_3)$ are edge-graceful, by providing their corresponding edge-graceful labels.

A Generalized core-satellite graph Θ(c, S, η∗) belongs to the family of graphs of diameter two. It has a central core of nodes connected to a few satellites, where all satellite cliques are not identical and might be of different sizes. These graphs can be used to model any real-world complex network. Using core-satellite graphs, properties like hierarchical structure can be conveniently modeled for large complex networks. In this paper, we obtain the lower and upper bounds for the spectral radius and signless Laplacian spectral radius of the generalized core-satellite graph, in terms of number of vertices, number of edges, and the graph parameters associated with the structure of the graph in both satellites and the core.

An edge coloring of a graph G refers to the process of assigning colors to the edges such that any two edges sharing a common vertex receive different colors. The Grundy edge chromatic number, denoted by (G), is defined as the largest number of colors that can be used by a greedy edge-coloring procedure across all permutations of the edges. This work focuses on computing (G), for the shadow graphs derived from families of graphs, including star graphs, double star graphs, comb graphs, fan graphs, and ladder graphs.

Graphs are useful for analyzing the structure models in computer science, operations research, and sociology. Also, different types of graph products have several applications in modeling, including those found in network analysis, communication protocols, organizational structures, diverse fields, network analysis and even chemistry. In this article, we introduce the concept of dominant strong metric dimension of graphs (DSMD for short) as a generalization of strong metric dimension and also dominant metric dimension. We determine the DSMD for several families of graphs that include for instance K m , n , the fan graph F 1 , n , the wheel graph W 1 , n and the helm graph H n . Additionally, we study this invariant under the join product, corona product of two graphs and the generalized join graphs. Moreover, this article investigates the concept of dominant strong metric dimensions for some algebraic graphs associated with rings, which is the comaximal graph. In fact, by using the results on DSMD of the generalized join graphs, the dominant strong metric dimension of the comaximal graph of the ring of integers modulo n is investigated. Through this exploration, besides considering various graph products, we aim to provide a comprehensive framework for analyzing commutative rings and their associated graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.