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The purpose of this paper is to extend the concept of energy of a graph with self-loops to partial complement of a graph. The partial complement of a graph G with respect to a set S denoted by G ⊕ S is the graph obtained by removing the edges of ⟨S⟩ and adding edges which are not in ⟨S⟩ in G. Let G be a graph of order n with σ self-loops, then the energy is defined as E(G) = .... Based on this, we introduce the concept of energy of partial complement of a graph G with self-loops. [ABSTRACT FROM AUTHOR]

Graph theory is a fundamental area that explores the properties and relationships present within graph structures. This paper explores generalized graph complements by examining edge partitioning, a novel approach that extends the traditional graph complement. In analogy to the vertex partition-based definitions of generalized complements, two distinct types of generalized complements for a graph emerge through edge partitioning: k'-complement and k'(i)- complement of a graph. k'-complement of a graph emerges as a versatile tool with applications across multiple domains, due to its maximum edge configuration linked with a Fibonacci polynomial and its role as a maximal outerplanar graph. The paper systematically examines theorems and fundamental properties governing the structural relationships between the original graph and its generalized complements. The characterizations of cycles and paths are also analysed in this study. [ABSTRACT FROM AUTHOR]

The topological indices are the numerical parameters of a graph that characterize the topology of a graph and are usually graph invariant. The topological indices are classified based on the properties of graphs. The degree distance index is the topological index which is calculated by counting the degrees and distance between the vertices. In this paper, the degree distance index of the connected thorn graph, the graph obtained by joining an edge between two connected graphs, and one vertex union of two connected graphs are calculated. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The adjacency matrix of G is a square matrix of order n, whose elements are 1, if the corresponding vertices are adjacent and 0, if the corresponding vertices are non-adjacent. Each row of the adjacency matrix is called as a string. The Hamming distance between two strings s1 and s2 is the number of positions at which their values differ. The sum of Hamming distances between each pair of vertices is the Hamming index of underlying graph. In this paper, the Hamming indices of different product graphs are obtained. [ABSTRACT FROM AUTHOR]

Albertson energy Albϵ(G) is defined as the sum of the absolute values of the Albertson eigenvalues of G. This paper computes the Albertson energy of various graph products, relating it to graph properties like order, degree, energy, or Albertson energy of the base graph. Our primary focus is on deriving formulas for the Albertson energy of the Cartesian, strong, and tensor products when one graph is regular. We also compute the Albertson energy of join, corona and hierarchical products of regular graphs. Furthermore, we calculate the Albertson energy of p-shadow and p-duplicate graphs. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph. A topological invariant named KG-Sombor index was introduced by V. R. Kulli, defined as KG = KG(G) = ∑ue√ d(u) ² + d(e) ², where ∑ ue indicates summation over vertices u ∈ V (G) and the edges e ∈ E(G) that are incident to u. In this paper, we extend the KG-Sombor index of simple graphs to graph with self loops. We study some properties of KG-Sombor eigenvalues and few bounds for KG-Sombor energy of graphs with self loops and KG-Sombor characteristic polynomial of graphs with self loops. [ABSTRACT FROM AUTHOR]

The modern world has brought everyone closer through digital communications. People are highly dependent on digital services. The quest for a more complex coding algorithm to prevent data from breaching is never-ending. The more intricate the algorithm is, the safer the communication will be. So, the vogue is to find the most convoluted algorithm to provide secure communication. In this paper, a unique algorithm is developed using graph labeling, a complement of a graph and generalized complement. The algorithm generates two labeled graphs satisfying vertex even mean and vertex odd mean labelings. Encryption involves the process of complementation and combining both graphs by k-complementation of graphs to obtain a cipher graph. The reverse process is applied for decryption which involves splitting the cipher graph into two subgraphs by applying k-complement for the specified partition of the vertex set, taking the complement of the obtained graph/s and getting the values of the plaintext using the graph labeling method. The proposed algorithm is designed in such a way that it should be useful for all kinds of plaintexts even with special characters. To illustrate this, an app is developed in the Android platform for communication of messages using endto-end encryption. [ABSTRACT FROM AUTHOR]

Nowadays, the use of digital technology for communication is constantly growing. As a result, algorithms that provide more secure communication must be employed. Despite the fact that there are numerous encryption algorithms, app developers are constantly looking for a new or updated versions of the algorithms. In this proposed work, the vertex odd mean labeling method is used to produce edge values for a graph that serve as the encoded numbers for the characters of a plaintext. The complement of that graph is taken to generate a cipher graph. Communication will be safer since each text will have a distinct cipher graph. The text is encrypted and decrypted using the symmetric key technique. This algorithmic approach is useful for all texts with or without special characters. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced minimum covering maximum reverse degree energy of simple graphs. Few properties on minimum covering maximum reverse degree eigenvalues and bounds for minimum covering maximum reverse degree energy of a graph are achieved. Further minimum covering maximum reverse degree energy of some families of graphs are computed. [ABSTRACT FROM AUTHOR]

The energy E(G) of a graph G, defined as the sum of the absolute values of its eigenvalues, belongs to the most popular graph invariants in chemical graph theory. It originates from the π-electron energy in the Huckel molecular orbital model, but has also gained purely mathematical interest. Let q1, q2, . . ., qn be the signless Laplacian eigenvalues of G. The signless Laplacian energy of G has recently been defined as QE(G) = Pn i=1 |qi - 2m n |. In this paper, we define signless Laplacian energy of partial complements of a graph. Signless Laplacian spectrum of partial complements of the few classes of graphs are established. Some bounds and properties of signless Laplacian energy are obtained. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacency matrix of a graph G. Let s (v) denote the row entries of A (G) corresponding to the vertex v of G. The Hamming distance between the strings s (u) and s (v) is the number of positions in which s (u) and s (v) differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of G ⊕ S. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V. A set D - V is a dominating set of G if each vertex of V - D is adjacent to at least one vertex of D. The k (k(i))- complement of G is obtained by partitioning V into k partites and removing the edges between the vertices of different (same) partites in G and adding the edges between the vertices of different (same) partites which are not in G. This paper studies different domination numbers of k and k(i) complements of graphs. [ABSTRACT FROM AUTHOR]

Let P = { V 1 , V 2 , ... , V k } be a partition of vertex set V of order k ≥ 2 of a graph G (V , E). The k -complement of G denoted by G k P is defined as for all V i and V j in P , i ≠ j , remove the edges between V i and V j in G and add the edges between V i and V j which are not in G. The graph G is called k -self-complementary if G ≅ G k P . For a graph G , k (i) -complement of G denoted by G k (i) P is defined as for each V r remove the edges of G inside V r and add the edges of G ¯ by joining the vertices of V r . The graph G is called k (i) -self-complementary if G ≅ G k (i) P for some partition P of order k. In this paper, we determine generalized self-complementary graphs of forest, double star and unicyclic graphs. [ABSTRACT FROM AUTHOR]

An assignment of distinct colors { 1 , 2 , 3 , ... , k } to the vertices u and v of a graph G (V , E) such that the distance between u and v is at most d is called d -distance coloring of G. Suppose C 1 , C 2 , ... , C k are the color classes of d -distance coloring and | | C i | − | C j | | ≤ 1 for any 1 ≤ i , j ≤ k , then G is d -distance equitable colored graph. In this paper, we obtain d -distance chromatic number and d -distance equitable chromatic number of graphs like P m ⊙ G , P m ¯ ⊙ G , P m ⊠ P n and P m ⊗ P n . [ABSTRACT FROM AUTHOR]

A dominating set for a graph G = (V, E) is a subset D of V such that every point not in D is adjacent to at least one member of D. Let P = {P1, P2,. .., P k} be a partition of point set V (G). For all Pi and Pj in P of order k ≥ 2, i ̸ = j, delete the lines between Pi and Pj in G and include the lines between Pi and Pj which are not in G. The resultant graph thus obtained is k-complement of G with respect to the partition P and is denoted by G P k. For each set Vr in P of order k ≥ 1, delete the lines of G inside Vr and insert the lines of G joining the points of Vr. The graph G P k(i) thus obtained is called the k(i)-complement of G with respect to the partition P. In this paper, we define D-complement and D(i)-complement of a graph G. Further we study various properties of D and D(i) complements of a given graph. Index Terms-D-complement, D(i)-complement, γ-complement and γ(i)-complement. [ABSTRACT FROM AUTHOR]

The color energy of a graph is defined as sum of absolute color eigenvalues of graph, denoted by Ec(G). Let Gc = (V,E) be a color graph and P = {V1, V2, . . ., Vk} be a partition of V of order k ≥ 1. The k -color complement {Gc}Pk of Gc is defined as follows: For all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj and add the edges which are not in Gc such that end vertices have different colors. For each set Vr in the partition P, remove the edges of Gc inside Vr, and add the edges of Gc (the complement of Gc ) joining the vertices of Vr. The graph {Gc}Pk(i) thus obtained is called the k(i)− color complement of Gc with respect to the partition P of V . In this paper, we compute color Laplacian energy of generalised complements of few standard graphs. Color Laplacian energy depends on assignment of colors to the vertices and the partition of V (G). [ABSTRACT FROM AUTHOR]

The color energy of a graph is defined as sum of absolute color eigenvalues of graph, denoted by Ec(G). Let Gc = (V, E) be a color graph and P = {V1, V2, ..., Vk} be a partition of V of order k ≥ 1. The k-color complement {Gc}kP of Gc is defined as follows: For all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj and add the edges which are not in Gc such that end vertices have different colors. For each set Vr in the partition P, remove the edges of Gc inside Vr, and add the edges of Gc (the complement of Gc) joining the vertices of Vr. The graph {Gc}Pk(iP) thus obtained is called the k(i) -- color complement of Gc with respect to the partition P of V. In this paper we characterize generalized color complements of some graphs. We also compute color energy of generalised complements of star, complete, complete bipartite, crown, cocktail party, double star and friendship graphs. [ABSTRACT FROM AUTHOR]

Let P = {V1, V2, V3, . . . , Vk} be a partition of vertex set V (G) of order k ≥ 2. For all Vi and Vj in P, i 6= j, remove the edges between Vi and Vj in graph G and add the edges between Vi and Vj which are not in G. The graph GP k thus obtained is called the k-complement of graph G with respect to a partition P. For each set Vr in P, remove the edges of graph G inside Vr and add the edges of G (the complement of G) joining the vertices of Vr. The graph GPk(i) thus obtained is called the k(i)-complement of graph G with respect to a partition P. In this paper, we study Laplacian energy of generalized complements of some families of graph. An effort is made to throw some light on showing variation in Laplacian energy due to changes in a partition of the graph. [ABSTRACT FROM AUTHOR]

Centrality measures are scalar values given to each node in the graph to quantify its importance based on an assumption. Stress and tension are the centrality index based on shortest path. In this paper, stress and tension of generalized complements of some standard graphs are calculated by counting the number of geodesics of different length. [ABSTRACT FROM AUTHOR]

Let G be a binary labeled graph and Al (G) = (lij) be its label adjacency matrix. For a vertex vi, we define label degree as ... . In this paper, we define label Laplacian energy LEl (G). It depends on the underlying graph G and labels of the vertices. We compute label Laplacian spectrum of families of graph. We also obtain some bounds for label Laplacian energy. [ABSTRACT FROM AUTHOR]

Let A(G) be the adjacency matrix of a graph G. Let s(vi) denote the row entries of A(G) corresponding to the vertex vi of G. The Hamming distance between the strings s(ui) and s(vi) is the number of positions in which their elements differ. The sum of Hamming distance between all the pairs of vertices is the Hamming index of a graph. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of various products of complete bipartite and complete graph. We also compute the Hamming index generated by the adjacency matrix of these graph products. [ABSTRACT FROM AUTHOR]

The partial complement of a graph G with respect to a set S denoted by G⊕S is the graph obtained by removing the edges of 〈S〉 and adding edges which are not in 〈S〉 in G. In this paper we introduce the concept of Seidel energy of partial complement of a graph. Some bounds are obtained for Seidel energy of partial complementary graph. We compute Seidel energy and Seidel spectrum for partial complement of several classes of graph. [ABSTRACT FROM AUTHOR]

Centrality measures are scalar values given to each vertex in the graph to quantify its importance based on an assumption. Stress and tension are the two centrality measures which depends on shortest paths. Stress of a vertex v is the number of shortest paths passing through vertex v whereas, tension on an edge e is the number of shortest paths passing through the edge e. In this paper, authors obtain formulae for the evaluation of stress and tension of certain graphs and graph operations with diameter less than or equal to two. Also obtain tension of some wheel related graphs such as gear graph, helm graph, flower graph and sunflower graph. [ABSTRACT FROM AUTHOR]