The paper begins with a simple circular lock problem that shows how the Combinatorial Nullstellensatz relates to the discrete Fourier Transform.Specifically, the lock shows a relationship between detecting perfect matchings in bipartite graphs using the Combinatorial Nullstellensatz and detecting a maximum rank independent set in the intersection of two matroids in the Fourier transform of a specially chosen function. Finally, an application of the uncertainity principle computes a lower bound for the product of perfect matchings and the number of independent sets.

A \({magic\; square}\) of order \(n\) is an \(n \times n\) array of integers from \(1, 2, \ldots, n^2\) such that the sum of the integers in each row, column, and diagonal is the same number. Two magic squares are \({equivalent}\) if one can be obtained from the other by rotation or reflection. The \({complement}\) of a magic square \(M\) of order \(n\) is obtained by replacing every entry \(a\) with \(n^2 + 1 – a\), yielding another magic square. A magic square is \({self-complementary}\) if it is equivalent to its complement. In this paper, we prove a structural theorem characterizing self-complementary magic squares and present a method for constructing self-complementary magic squares of even order. Combining this construction with the structural theorem and known results on magic squares, we establish the existence of self-complementary magic squares of order \(n\) for every \(n \geq 3\).

Let \(G\) be a graph on \(n\) vertices. If for any ordered set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\), where the vertices in \(S\) appear in the sequence order \(v_1, v_2, \ldots, v_k\), there exists a \(v_1-v_k\) (Hamiltonian) path containing \(S\) in the given order, then \(G\) is \(k\)-ordered (Hamiltonian) connected. In this paper, we show that if \(G\) is \((k+1)\)-connected and \(k\)-ordered connected, then for any ordered set \(S\), there exists a \(v_1-v_k\) path \(P\) containing \(S\) in the given order such that \(|P| \geq \min\{n, \sigma_2(G) – 1\}\), where \(\sigma_2(G) = \min\{d_G(u) + d_G(v) : u,v \in V(G); uv \notin E(G)\}\) when \(G\) is not complete, and \(\sigma_2(G) = \infty\) otherwise. Our result generalizes several related results known before.

Let \(G\) be a simple graph. The incidence energy ( \(IE\) for short ) of \(G\) is defined as the sum of the singular values of the incidence matrix. In this paper, a new lower bound for \(IE\) of graphs in terms of the maximum degree is given. Meanwhile, an upper bound and a lower bound for \(IE\) of the subdivision graph and the total graph of a regular graph \(G\) are obtained, respectively.

The Hosoya polynomial of a graph \(G\) with vertex set \(V(G)\) is defined as \(H(G, z) = \sum_{u,v \in V(G)} x^{d_G(u,v)}\), where \(d_G(u,v)\) is the distance between vertices \(u\) and \(v\). A toroidal polyhex \(H(p,q,t)\) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, described by a string \((p,q,t)\) of three integers \((p \geq 2, q \geq 1, 0 \leq t \leq p-1)\). In this paper, we derive an analytical formula for calculating the Hosoya polynomial of \(H(p,q,t)\) for \(t = 0\) or \(p\leq 2q\) or \(p \leq q+t\). Notably, some earlier results in [2, 6, 26] are direct corollaries of our main findings.

Kotani and Sunada introduced the oriented line graph as a tool in the study of the Ihara zeta function of a finite graph. The spectral properties of the adjacency operator on the oriented line graph can be linked to the Ramanujan condition of the graph. Here, we present a partial characterization of oriented line graphs in terms of forbidden subgraphs. We also give a Whitney-type result, as a special case of a result by Balof and Storm, establishing that if two graphs have the same oriented line graph, they are isomorphic.

Let \(A\) be the \((0,1)\)-adjacency matrix of a simple graph \(G\), and \(D\) be the diagonal matrix \(diag(d_1, d_2, \ldots, d_n)\), where \(d_i\) is the degree of the vertex \(v_i\). The matrix \(Q(G) = D + A\) is called the signless Laplacian of \(G\). In this paper, we characterize the extremal graph for which the least signless Laplacian eigenvalue attains its minimum among all non-bipartite unicyclic graphs with given order and diameter.

In this paper, we investigate some commutativity conditions and extend a remarkable result of Ram Awtar, when Lie ideal \(U\) becomes the part of the centre of \(M\) \(A\)-semiring \(R\).

A pebbling move involves removing two pebbles from one vertex and placing one on an adjacent vertex. The optimal pebbling number of a graph \(G\), denoted by \(f_{opt}(G)\), is the least positive integer \(n\) such that \(n\) pebbles are placed suitably on vertices of \(G\) and, for any specified vertex \(v\) of \(G\), one pebble can be moved to \(v\) through a sequence of pebbling moves. In this paper, we determine the optimal pebbling number of the square of paths and cycles.

In this paper, we verify the list edge coloring conjecture for pseudo- outerplanar graphs with maximum degree at least \(5\) and the equitable \(\Delta\)-coloring conjecture for all pseudo-outerplanar graphs.

We prove that the Cartesian product of two directed cycles of lengths \(n_1\) and \(n_2\) contains an antidirected Hamilton cycle, and hence is decomposable into antidirected Hamilton cycles, if and only if \(\gcd(n_1, n_2) = 2\). For the Cartesian product of \(k > 2\) directed cycles, we establish new sufficient conditions for the existence of an antidirected Hamilton cycle.

Let \(T\) be a tree with no vertices of degree \(2\) and at least one vertex of degree \(3\) or more. A Halin graph \(G\) is a plane graph obtained by connecting the leaves of \(T\) in the cyclic order determined by the planar drawing of \(T\). Let \(\Delta\), \(\lambda(G)\), and \(\chi(G^2)\) denote, respectively, the maximum degree, the \(L(2,1)\)-labeling number, and the chromatic number of the square of \(G\). In this paper, we prove the following results for any Halin graph \(G\): (1) \(\chi(G^2) \leq \Delta + 3\), and moreover \(\chi(G^2) = \Delta + 1\) if \(\Delta \geq 6\); (2) \(\lambda(G) \leq \Delta + 7\), and moreover \(\lambda(G) \leq \Delta + 2\) if \(\Delta \geq 9\).

In this paper, we investigate the zero divisor graph \(G_I(P)\) of a poset \(P\) with respect to a semi-ideal \(I\). We show that the girth of \(G_I(P)\) is \(3\), \(4\), or \(\infty\). In addition, it is shown that the diameter of such a graph is either \(1\), \(2\), or \(3\). Moreover, we investigate the properties of a cut vertex in \(G_I(P)\) and study the relation between semi-ideal \(I\) and the graph \(G_I(P)\), as established in (Theorem 3.9).

A graph \(G\) is \({super-connected}\), or \({super-\(\kappa\)}\), if every minimum vertex-cut isolates a vertex of \(G\). Similarly, \(G\) is \({super-restricted \;edge-connected}\), or \({super-\(\lambda’\)}\), if every minimum restricted edge-cut isolates an edge. We consider the total graph \(T(G)\) of \(G\), which is formed by combining the disjoint union of \(G\) and the line graph \(L(G)\) with the lines of the subdivision graph \(S(G)\); for each line \(l = (u,v)\) in \(G\), there are two lines in \(S(G)\), namely \((l,u)\) and \((l,v)\). In this paper, we prove that \(T(G)\) is super-\(\kappa\) if \(G\) is super-\(\kappa\) graph with \(\delta(G) \geq 4\). \(T(G)\) is super-\(\lambda’\) if \(G\) is \(k\)-regular with \(\kappa(G) \geq 3\). Furthermore, we provide examples demonstrating that these results are best possible.

The paper construct infinite classes of non-isomorphic \(3\)-connected simple graphs with the same total genus polynomial, using overlap matrix, symmetry and Gustin representation. This answers a problem (Problem \(3\) of Page \(38\)) of L.A. McGeoch in his PHD thesis.
The result is helpful for firms to make marketing decisions by calculating the graphs of user demand relationships of different complex ecosystems of platform products and comparing genus polynomials.

A necessary and sufficient condition of the complement to be cordial and its application are obtained.

In this paper, we introduce the notion of blockwise-bursts in array codes equippped with m-metric \([13]\) and obtain some bounds on the parameters of $m$-metric array codes for the detection and correction of blockwise-burst array errors.

Let \(G\) be a graph, and let \(a\) and \(b\) be integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(G)\). In this paper, we obtain a sufficient condition for a graph to have \([a, b]\)-factors including given edges, extending a well-known sufficient condition for the existence of a \(k\)-factor.

We introduce the domination polynomial of a graph \(G\). The domination polynomial of a graph \(G\) of order \(n\) is defined as \(D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma(G)\) is the domination number of \(G\). We obtain some properties of \(D(G, x)\) and its coefficients, and compute this polynomial for specific graphs.

For a tree \(T\), \(Leaf(T)\) denotes the set of leaves of \(T\), and \(T – Leaf(T)\) is called the stem of \(T\). For a graph \(G\) and a positive integer \(m\), \(\sigma_m(G)\) denotes the minimum degree sum of \(m\) independent vertices of \(G\). We prove the following theorem: Let \(G\) be a connected graph and \(k \geq 2\) be an integer. If \(\sigma_3(G) \geq |G| – 2k + 1\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves.

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most \(1\). The equitable chromatic threshold of a graph \(G\), denoted by \(\chi_m^*(G)\), is the minimum \(k\) such that \(G\) is equitably \(k’\)-colorable for all \(k’ > k\). Let \(G \times H\) denote the direct product of graphs \(G\) and \(H\). For \(n \geq m \geq 2\), we prove that \(\chi_m^*(K_m \times K_n)\) equals \(\left\lceil \frac{mn}{m+1} \right\rceil\) if \(n \equiv 2, \ldots, m \pmod{m+1}\), and equals \(m\left\lceil \frac{n}{s^*} \right\rceil\) if \(n \equiv 0, 1 \pmod{m+1}\), where \(s^*\) is the minimum positive integer such that \(s^* \nmid n\) and \(s^* \geq m+2\).

For an undirected graph \(G\) and a natural number \(n\), a \(G\)-design of order \(n\) is an edge partition of the complete graph \(K_n\) with \(n\) vertices into subgraphs \(G_1, G_2, \ldots\), each isomorphic to \(G\). A set \(T \subset V(K_n)\) is called a blocking set if it intersects the vertex set \(V(G_i)\) of each \(G_i\) in the decomposition but contains none of them. Extending previous work [J. Combin. Designs \(4 (1996), 135-142]\), where the authors proved that cycle designs admit no blocking sets, we establish that this result holds for all graphs \(G\). Furthermore, we show that for every graph \(G\) and every integer \(k \geq 2\), there exists a non-\(k\)-colorable \(G\)-design.

Let \(G\) be a planar graph with maximum degree \(\Delta(G)\). The least integer \(k\) such that \(G\) can be partitioned into \(k\) edge-disjoint forests, where each component is a path of length at most \(2\), is called the linear \(2\)-arboricity of \(G\), denoted by \(la_2(G)\). We establish new upper bounds for the linear \(2\)-arboricity of certain planar graphs.

A graph \(G\) of order \(n\) is called a bicyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n+ 1\). In this paper, we study the lexicographic ordering of bicyclic graphs by spectral moments. For each of the three basic types of bicyclic graphs on a fixed number of vertices maximal and minimal graphs in the mentioned order are determined.

An edge irregular total \(k\)-labeling of a graph \(G = (V, E)\) is a labeling \(f: V \cup E \to \{1, 2, \ldots, k\}\) such that the total edge-weights \(wt(xy) = f(x) + f(xy) + f(y)\) are distinct for all pairs of distinct edges. The minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength of \(G\). In this paper, we determine the exact value of the total edge irregularity strength of the Cartesian product of two paths \(P_n\) and \(P_m\). Our result provides further evidence supporting a recent conjecture of Ivančo and Jendrol.

For a vertex set \(S\) with cardinality at least \(2\) in a graph \(G\), a tree connecting \(S\), known as a Steiner tree or \(S\)-tree, is required. Two \(S\)-trees \(T\) and \(T’\) are internally disjoint if \(V(T) \cap V(T’) = S\) and \(E(T) \cap E(T’) = \emptyset\). Let \(\kappa_G(G)\) denote the maximum number of internally disjoint Steiner trees connecting \(S\) in \(G\). The generalized \(k\)-connectivity \(\kappa_k(G)\) of \(G\), introduced by Chartrand et al., is defined as \(\min_{S \subseteq V(G), |S|=k} \kappa_G(S)\). This paper establishes a sharp upper bound for generalized \(k\)-connectivity. Furthermore, graphs of order \(n\) with \(\kappa_3(G) = n-2,n-3\) are characterized.

A hypergraph \(\mathcal{H}\) is said to be \(p\)-Helly when every \(p\)-wise intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has nonempty total intersection. Such hypergraphs were characterized by Berge and Duchet in 1975, and since then they have appeared in various contexts, particularly for \(p=2\), where they are known as Helly hypergraphs. An interesting generalization due to Voloshin considers both the number of intersecting sets and their intersection sizes: a hypergraph \(\mathcal{H}\) is \((p,q,s)\)-Helly if every \(p\)-wise \(q\)-intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has total intersection of cardinality at least \(s\). This work proposes a characterization for \((p,q,s)\)-Helly hypergraphs, leading to an efficient algorithm for recognizing such hypergraphs when \(p\) and \(q\) are fixed parameters.

A \(k\)-chromatic graph \(G\) is \(uniquely\) \(k\)-\(colorable\) if \(G\) has only one \(k\)-coloring up to permutation of the colors. In this paper, we focus on uniquely \(k\)-colorable graphs on surfaces. Let \({F}^2\) be a closed surface, excluding the sphere, and let \(\chi({F}^2)\) denote the maximum chromatic number of graphs embeddable on \({F}^2\). We shall prove that the number of uniquely \(k\)-colorable graphs on \({F}^2\) is finite if \(k \geq 5\), and characterize uniquely \(\chi({F}^2)\)-colorable graphs on \({F}^2\). Moreover, we completely determine uniquely \(k\)-colorable graphs on the projective plane for \(k \geq 5\).

Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of a graph, denoted by \(f(G)\), is the minimal integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we calculate the pebbling number of the graph \(D_{n,C_m}\) and consider the relationship the pebbling number between the graph \(D_{n,C_m}\) and the subgraphs of \(D_{n,C_m}\).

Let \(G\) and \(H\) be two graphs. A proper vertex coloring of \(G\) is called a dynamic coloring if, for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic coloring with \(k\) colors is denoted by \(\chi_2(G)\). We denote the Cartesian product of \(G\) and \(H\) by \(G \square H\). In this paper, we prove that if \(G\) and \(H\) are two graphs and \(\delta(G) \geq 2\), then \(\chi_2(G \square H) \leq \max(\chi_2(G), \chi(H))\). We show that for every two natural numbers \(m\) and \(n\), \(m, n \geq 2\), \(\chi_2(P_m \square P_n) = 4\). Additionally, among other results, it is shown that if \(3\mid mn\), then \(\chi_2(C_m \square C_n) = 3\), and otherwise \(\chi_2(C_m \square C_n) = 4\).

In \([1]\), Hosam Abdo and Darko Dimitrov introduced the total irregularity of a graph. For a graph \(G\), it is defined as
\[\text{irr}_t(G) =\frac{1}{2} \sum_{{u,v} \in V(G)} |d_G(u) – d_G(v)|,\]
where \(d_G(u)\) denotes the vertex degree of a vertex \(u \in V(G)\). In this paper, we introduce two transformations to study the total irregularity of unicyclic graphs and determine the graph with the maximal total irregularity among all unicyclic graphs with \(n\) vertices.

We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, \(et \;al\). The solution to the original problem is the well known Catalan numbers, while the variations discussed in this paper yield the Motzkin numbers and other related sequences. For this variation, we present a generating function for the sum of the labels on the balls.

A graph \(G\) of order \(n\) is called a tricyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n + 2\). Let \(\mathcal{T}_n\) denote the set of all tricyclic graphs on \(n\) vertices. In this paper, we determine the first to nineteenth largest Laplacian spectral radii among all graphs in the class \(\mathcal{T}_n\) (for \(n \geq 11\)), together with the corresponding graphs.

The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we determine the lower bounds for the Hosoya index of unicyclic graph with a given diameter. The corresponding extrenal graphs are characterized.

A subset \(S\) of vertices of a graph \(G\) is called a global connected dominating set if \(S\) is both a global dominating set and a connected dominating set. The global connected domination number, denoted by \(\gamma_{gc}(G)\), is the minimum cardinality of a global connected dominating set of \(G\). In this paper, sharp bounds for \(\gamma_{gc}\) are supplied, and all graphs attaining those bounds are characterized. We also characterize all graphs of order \(n\) with \(\gamma_{gc} = k\), where \(3 \leq k \leq n-1\). Exact values of this number for trees and cycles are presented as well.

Let \(\mathbb{F}_q^n\) denote the \(n\)-dimensional row vector space over the finite field \(\mathbb{F}_q\), where \(n \geq 2\). An \(l\)-partial linear map of \(\mathbb{F}_q^n\) is a pair \((V, f)\), where \(V\) is an \(l\)-dimensional subspace of \(\mathbb{F}_q^n\) and \(f: V \to \mathbb{F}_q^n\) is a linear map. Let \(\mathcal{L}\) be the set of all partial linear maps of \(\mathbb{F}_q^n\) containing \(1\). Ordered \(\mathcal{L}\) by ordinary and reverse inclusion, two families of finite posets are obtained. This paper proves that these posets are lattices, discusses their geometricity, and computes their characteristic polynomials.

A total coloring of a graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring \(h\) of a simple graph \(G = (V, E)\) is a proper total coloring of \(G\) such that \(H(u) \neq H(v)\) for any two adjacent vertices \(u\) and \(v\), where \(H(u) = \{h(wu) \mid wu \in E(G)\} \cup \{h(u)\}\) and \(H(v) = \{h(xv) \mid xv \in E(G)\} \cup \{h(v)\}\). The minimum number of colors required for a proper total coloring (resp. an adjacent vertex-distinguishing total coloring) of \(G\) is called the total chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of \(G\) and denoted by \(\chi_t(G)\) (resp. \(\chi_{at}(G)\)). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\), \(\chi(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2\). \(G\) is called Type 1 (resp. Type 2) if \(\chi_t(G) = \Delta(G) + 1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the augmented cube \(AQ_n\) is of Type 1 for \(n \geq 4\). We also consider the adjacent vertex-distinguishing total chromatic number of \(AQ_n\) and prove that \(\chi_{at}(AQ_n) = \Delta(AQ_n) + 2\) for \(n \geq 3 \).

The Channel Assignment Problem is often modeled by integer vertex-labelings of graphs. We will examine \(L(2,1)\)-labelings that realize the span \(\lambda\) of a simple, connected graph \(G = (V, E)\). We define the utility of \(G\) to be the number of possible expansions that can occur on \(G\), where an expansion refers to an opportunity to add a new vertex \(u\) to \(G\), with label \(\lambda(u)\), such that:

1. edges are added between \(u\) and \(v\);
2. edges are added between \(u\) and the neighbors of \(v\); and
3. the resulting labeling of the graph is a valid \(L(2, 1)\)-labeling.

Building upon results of Griggs, Jin, and Yeh, we use known values of \(\lambda\) to compute utility for several infinite families and analyze the utility of specific graphs that are of interest elsewhere.

A Sidon set \(S\) is a set of integers where the number of solutions to any integer equation \(k = k_1 + k_2\) with \(k_1, k_2 \in S\) is at most \(2\). If \(g \geq 2\), the set \(S\) is a generalized Sidon set. We consider Sidon sets modulo \(n\), where the solutions to addition of elements are considered under a given modulus. In this note, we give a construction of a generalized Sidon set modulo \(n\) from any known Sidon set.

In an ordered graph \(G\), a set of vertices \(S\) with a pre-coloring of the vertices of \(S\) is said to be a greedy defining set (GDS) if the greedy coloring of \(G\) with fixed colors of \(S\) yields a \(\chi(G)\)-coloring of \(G\). This concept first appeared in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. The smallest size of any GDS in a graph \(G\) is called the greedy defining number of \(G\). We show that determining the greedy defining number of bipartite graphs is an NP-complete problem, affirmatively answering a problem mentioned in a previous paper. Additionally, we demonstrate that this number for forests can be determined in linear time. Furthermore, we present a method for obtaining greedy defining sets in Latin squares and, using this method, show that any \(n \times n\) Latin square has a GDS of size at most \(n^2 – (n \log 4n)/4\).

Multi-receiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message. In this paper, we construct one multi-receiver authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of this codes are also computed.

Resistance distance was introduced by Klein and Randic as a generalization of the classical distance. The Kirchhoff index \(Kf(G)\) of a graph \(G\) is the sum of resistance distances between all pairs of vertices. In this paper, we determine the bicyclic graph of order \(n \geq 8\) with maximal Kirchhoff index. This improves and extends an earlier result by Zhang \(et\; al. [19]\).

Bereg and Wang defined a new class of highly balanced \(d\)-ary trees which they call \(k\)-trees; these trees have the interesting property that the internal path length and thus the Wiener index can be calculated quite easily. A \(k\)-tree is characterized by the property that all levels, except for the last \(k\) levels, are completely filled. Bereg and Wang claim that the number of \(k\)-trees is exponentially increasing, but do not give an asymptotic formula for it. In this paper, we study the number of \(d\)-ary \(k\)-trees and the number of mutually non-isomorphic \(d\)-ary \(k\)-trees, making use of a technique due to Flajolet and Odlyzko.

A group \(G\) is said to be a \(B_k\)-group if for any \(k\)-subset \(\{a_1, \ldots, a_k\}\) of \(G\), \(\left|\{a_ia_j \mid 1 \leq i, j \leq k\}\right| \leq \frac{k(k+1)}{2}\). In this paper, a complete classification of \(B_5\)-groups is given.

The local-restricted-edge-connectivity \(\lambda'(e, f)\) of two nonadjacent edges \(e\) and \(f\) in a graph \(G\) is the maximum number of edge-disjoint \(e\)-\(f\) paths in \(G\). It is clear that \(\lambda'(G) = \min\{\lambda'(e, f) \mid e \text{ and } f \text{ are nonadjacent edges in } G\}\), and \(\lambda'(e, f) \leq \min\{\xi(e), \xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\), where \(\lambda(G)\), \(\xi(e)\), and \(\xi(f)\) denote the restricted-edge-connectivity of \(G\), the edge-degree of edges \(e\) and \(f\), respectively. Let \(\xi(G)\) be the minimum edge-degree of \(G\). We call a graph \(G\) optimally restricted-edge-connected when \(\lambda'(G) = \xi(G)\) and optimally local-restricted-edge-connected if \(\lambda'(e, f) = \min\{\xi(e),\xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\). In this paper, we show that some known sufficient conditions that guarantee that a graph is optimally restricted-edge-connected also guarantee that it is optimally local-restricted-edge-connected.